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  • 1.
    Becker, Roland
    et al.
    Université de Pau.
    Burman, Erik
    University of Sussex.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A finite element time relaxation method2011In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 349, no 5-6, p. 353-356Article in journal (Refereed)
    Abstract [en]

    We discuss a finite element time-relaxation method for high Reynolds number flows. The method uses local projections on polynomials defined on macroelements of each pair of two elements sharing a face. We prove that this method shares the optimal stability and convergence properties of the continuous interior penalty (CIP) method. We give the formulation both for the scalar convection-diffusion equation and the time-dependent incompressible Euler equations and the associated convergence results. This note finishes with some numerical illustrations.

  • 2.
    Becker, Roland
    et al.
    Université de Pau et des Pays de l'Adour.
    Burman, Erik
    University of Sussex Falmer.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A hierarchical NXFEM for fictitious domain simulations2011In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 86, no 4-5, p. 549-559Article in journal (Refereed)
    Abstract [en]

    We suggest a fictitious domain method, based on the Nitsche XFEM method of (Comput. Meth. Appl. Mech. Engrg 2002; 191: 5537-5552), that employs a band of elements adjacent to the boundary. In contrast, the classical fictitious domain method uses Lagrange multipliers on a line (surface) where the boundary condition is to be enforced. The idea can be seen as an extension of the Chimera method of (ESAIM: Math. Model Numer. Anal. 2003; 37: 495-514), but with a hierarchical representation of the discontinuous solution field. The hierarchical formulation is better suited for moving fictitious boundaries since the stiffness matrix of the underlying structured mesh can be retained during the computations. Our technique allows for optimal convergence properties irrespective of the order of the underlying finite element method.

  • 3.
    Becker, Roland
    et al.
    Université de Pau et des Pays de l’Adour.
    Burman, Erik
    University of Sussex.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity2009In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 198, no 41-44, p. 3352-3360Article in journal (Refereed)
    Abstract [en]

    In this note we propose a finite element method for incompressible (or compressible) elasticity problems with discontinuous modulus of elasticity (or, if compressible, Poisson's ratio). The problem is written on mixed form using P(1)-continuous displacements and elementwise P(0) pressures, leading to the possibility of eliminating the pressure beforehand in the compressible case. In the incompressible case, the method is augmented by a stabilization term, penalizing the pressure jumps. We show a priori error estimates under certain regularity hypothesis. In particular we prove that if the exact solution is sufficiently smooth in each subdomain then the convergence order is optimal.

  • 4.
    Becker, Roland
    et al.
    Université de Pau et des Pays de l'Adour.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A simple pressure stabilization method for the Stokes equation2008In: Communications in Numerical Methods in Engineering, ISSN 1069-8299, E-ISSN 1099-0887, Vol. 24, no 11, p. 1421-1430Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider a stabilization method for the Stokes problem, using equal-order interpolation of the pressure and velocity, which avoids the use of the mesh size parameter in the stabilization term. We show that our approach is stable for equal-order interpolation in the case of piecewise linear and piecewise quadratic polynomials on triangles. In the case of linear polynomials, we retrieve a well-known idea of using mass lumping as a stabilization mechanism.

  • 5. Becker, Roland
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Discontinuous Galerkin methods for convection–diffusion problems with arbitrary Péclet number2000In: Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2003 / [ed] P. Neittaanmäki, T. Tiihonen, and P. Tarvainen, Berlin: Springer , 2000Conference paper (Refereed)
  • 6.
    Becker, Roland
    et al.
    Universität Heidelberg.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G
    Chalmers University of Technology.
    Energy norm a posteriori error estimation for discontinuous Galerkin methods2003In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 192, no 5-6, p. 723-733Article in journal (Refereed)
    Abstract [en]

    In this paper we present a residual-based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous Galerkin approximations of elliptic problems. The theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed. We also present some illustrating numerical examples.

  • 7.
    Becker, Roland
    et al.
    University of Heidelberg.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Stenberg, Rolf
    Helsinki University of Technology.
    A finite element method for domain decomposition with non-matching grids2003In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 37, no 2, p. 209-225Article in journal (Refereed)
    Abstract [en]

    In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

  • 8.
    Burman, Erik
    et al.
    University College London, Gower Street, UK.
    Elfverson, Daniel
    Umeå University, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Umeå University, Sweden.
    Larsson, Karl
    Umeå University, Sweden.
    A cut finite element method for the Bernoulli free boundary value problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, p. 598-618Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H1norm. Finally, we present illustrating numerical results.

  • 9.
    Burman, Erik
    et al.
    University College London, London, United Kingdom.
    Elfverson, Daniel
    Umeå University, Umeå, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå University, Umeå, Sweden.
    Larsson, Karl
    Umeå University, Umeå, Sweden.
    Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 462-479Article in journal (Refereed)
    Abstract [en]

    We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represent parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche's method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero. 

  • 10.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, Gower Street, London, United Kingdom.
    Elfverson, Daniel
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Shape and topology optimization using CutFEM2017In: Simulation for Additive Manufacturing 2017, Sinam 2017, International Center for Numerical Methods in Engineering (CIMNE), 2017, p. 208-209Conference paper (Refereed)
    Abstract [en]

    We present a shape and topology optimization method based on the cut finite element method, see [1],[2], and [3], for the optimal compliance problem in linear elasticity and problems involving restrictionson the stresses.The elastic domain is defined by a level-set function, and the evolution of the domain is obtained bymoving the level-set along a velocity field using a transport equation. The velocity field is defined tobe the largest decreasing direction of the shape derivative that resides in a certain Hilbert space and iscomputed by solving an elliptic problem, associated with the bilinear form in the Hilbert space, with theshape derivative as right hand side. The velocity field may thus be viewed as the Riesz representationof the shape derivative on the chosen Hilbert space.We thus obtain a coupled problem involving three partial differential equations: (1) the elasticity problem,(2) the elliptic problem that determines the velocity field, and (3) the transport problem for thelevelset function. The elasticity problem is solved using a cut finite element method on a fixed backgroundmesh, which completely avoids re–meshing when the domain is updated. The levelset functionand the velocity field is approximated by standard conforming elements on the background mesh. Wealso employ higher order cut approximations including isogeometric analysis for the elasticity problem.In this case the levelset function and the velocity field are represented using linear elements on a refinedmesh in order to simplify the geometric and quadrature computations on the cut elements. To obtain astable method, stabilization terms are added in the vicinity of the cut elements at the boundary, whichprovides control of the variation of the solution in the vicinity of the boundary. We present numericalexamples illustrating the performance of the method.We also study an anisotropic material model that accounts for the orientation of the layers in an additivemanufacturing process and by including the orientation in the optimization problem we determine theoptimal choice of orientation.We present numerical results including test problems and engineering applications in additive manufacturing.

    References

    [1] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: discretizing geometryand partial differential equations. Internat. J. Numer. Methods Engrg., 104(7):472–501, 2015.

    [2] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. Shape optimization using thecut finite element method. Technical report, 2016. arXiv:1611.05673.

    [3] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. A cut finite element method forthe Bernoulli free boundary value problem. Comput. Methods Appl. Mech. Engrg., 317:598–618,2017.

  • 11.
    Burman, Erik
    et al.
    University College London, United Kingdom.
    Elfverson, Daniel
    Umeå universitet, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats
    Umeå universitet, Sweden.
    Larsson, Karl
    Umeå universitet, Sweden.
    Shape optimization using the cut finite element method2018In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 242-261Article in journal (Refereed)
    Abstract [en]

    We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity field using a transport equation. The velocity field is the largest decreasing direction of the shape derivative that satisfies a certain regularity requirement and the computation of the shape derivative is based on a volume formulation. Using the cut finite element method no re-meshing is required when updating the domain and we may also use higher order finite element approximations. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements at the boundary, which provides control of the variation of the solution in the vicinity of the boundary. We implement and illustrate the performance of the method in the two-dimensional case, considering both triangular and quadrilateral meshes as well as finite element spaces of different order.

  • 12. Burman, Erik
    et al.
    Fernandez, Miguel A.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Continuous interior penalty finite element method for Oseen's equations2006In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 44, no 3, p. 1248-1274Article in journal (Refereed)
    Abstract [en]

    In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [ Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207 - 216] to Oseen's equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/ velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L-2-control of the jump of the gradient over element faces ( edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.

  • 13.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A stabilized non-conforming finite element method for incompressible flow2006In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, no 23-24, p. 2881-2899Article in journal (Refereed)
    Abstract [en]

    In this paper we extend the recently introduced edge stabilization method to the case of non-conforming finite element approximations of the linearized Navier-Stokes equation. To get stability also in the convective dominated regime we add a term giving L-2-control of the jump in the gradient over element boundaries. An a priori error estimate that is uniform in the Reynolds number is proved and some numerical examples are presented.

  • 14.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A unified stabilized method for Stokes' and Darcy's equations2007In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 198, no 1, p. 35-51Article in journal (Refereed)
    Abstract [en]

    We use the lowest possible approximation order, piecewise linear, continuous velocities and piecewise constant pressures to compute solutions to Stokes equation and Darcy's equation, applying an edge stabilization term to avoid locking. We prove that the formulation satisfies the discrete inf-sup condition, we prove an optimal a priori error estimate for both problems. The formulation is then extended to the coupled case using a Nitsche-type weak formulation allowing for different meshes in the two subdomains. Finally, we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach.

  • 15.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom..
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Deriving Robust Unfitted Finite Element Methods from Augmented Lagrangian Formulations2017In: Geometrically Unfitted Finite Element Methods and Applications / [ed] Bordas, Stéphane P. A.; Burman, Erik; Larson, Mats G.; Olshanskii, Maxim A., Cham: Springer International Publishing , 2017, p. 1-24Conference paper (Refereed)
    Abstract [en]

    In this paper we will discuss different coupling methods suitable for use in the framework of the recently introduced CutFEM paradigm, cf. Burman et al. (Int. J. Numer. Methods Eng. 104(7):472–501, 2015). In particular we will consider mortaring using Lagrange multipliers on the one hand and Nitsche’s method on the other. For simplicity we will first discuss these methods in the setting of uncut meshes, and end with some comments on the extension to CutFEM. We will, for comparison, discuss some different types of problems such as high contrast problems and problems with stiff coupling or adhesive contact. We will review some of the existing methods for these problems and propose some alternative methods resulting from crossovers from the Lagrange multiplier framework to Nitsche’s method and vice versa.

  • 16.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems2004In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 193, no 15-16, p. 1437-1453Article in journal (Refereed)
    Abstract [en]

    In this paper we recall a stabilization technique for finite element methods for convection-diffusion-reaction equations, originally proposed by Douglas and Dupont [Computing Methods in Applied Sciences, Springer-Verlag, Berlin, 1976]. The method uses least square stabilization of the gradient jumps a across element boundaries. We prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates. We address the question of monotonicity of discrete Solutions and present some numerical examples illustrating the theoretical results.

  • 17.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Edge stabilization for the generalized Stokes problem: A continuous interior penalty method2006In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, no 19-22, p. 2393-2410Article in journal (Refereed)
    Abstract [en]

    In this note we introduce and analyze a stabilized finite element method for the generalized Stokes equation. Stability is obtained by adding a least squares penalization of the gradient jumps across element boundaries. The method can be seen as a higher order version of the Brezzi-Pitkdranta penalty stabilization [F. Brezzi, J. Pitkaranta, On the stabilization of finite element approximations of the Stokes equations, in: W. Hackbusch (Ed.), Efficient Solution of Elliptic Systems, Vieweg, 1984], but gives better resolution on the boundary for the Stokes equation than does classical Galerkin least-squares formulation. We prove optimal and quasi-optimal convergence properties for Stokes' problem and for the porous media models of Darcy and Brinkman. Some numerical examples are given.

  • 18.
    Burman, Erik
    et al.
    University of Sussex.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method2010In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 199, no 41-44, p. 2680-2686Article in journal (Refereed)
    Abstract [en]

    We propose a fictitious domain method where the mesh is cut by the boundary. The primal solution is computed only up to the boundary; the solution itself is defined also by nodes outside the domain, but the weak finite element form only involves those parts of the elements that are located inside the domain. The multipliers are defined as being element-wise constant on the whole (including the extension) of the cut elements in the mesh defining the primal variable. Inf-sup stability is obtained by penalizing the jump of the multiplier over element faces. We consider the case of a polygonal domain with possibly curved boundaries. The method has optimal convergence properties.

  • 19.
    Burman, Erik
    et al.
    Department of Mathematics, University of Sussex.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method2012In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, no 4, p. 328-341Article in journal (Refereed)
    Abstract [en]

    We extend the classical Nitsche type weak boundary conditions to a fictitious domain setting. An additional penalty term, acting on the jumps of the gradients over element faces in the interface zone, is added to ensure that the conditioning of the matrix is independent of how the boundary cuts the mesh. Optimal a priori error estimates in the H1- and L2-norms are proved as well as an upper bound on the condition number of the system matrix. 

  • 20. Burman, Erik
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem2011Report (Other academic)
    Abstract [en]

    We extend our results on fictitious domain methods for Poisson’s problem to the case of incompressible elasticity, or Stokes’ problem. The mesh is not fitted to the domain boundary. Instead boundary conditions are imposed using a stabilized Nitsche type approach. Control of the non-physical degrees of freedom, i.e., those outside the physical domain, is obtained thanks to a ghost penalty term for both velocities and pressures. Both inf–sup stable and stabilized velocity pressure pairs are considered.

  • 21.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem2014In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 48, no 3, p. 859-874Article in journal (Refereed)
    Abstract [en]

    We extend the classical Nitsche type weak boundary conditions to a fictitious domain setting. An additional penalty term, acting on the jumps of the gradients over element faces in the interface zone, is added to ensure that the conditioning of the matrix is independent of how the boundary cuts the mesh. Optimal a priori error estimates in the H 1- and L 2-norms are proved as well as an upper bound on the condition number of the system matrix.

  • 22.
    Burman, Erik
    et al.
    Chalmers University of Technology and Göteborg University.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems2010In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 30, no 3, p. 870-885Article in journal (Refereed)
    Abstract [en]

    In this paper we propose a class of jump-stabilized Lagrange multiplier methods for the finite-element solution of multidomain elliptic partial differential equations using piecewise-constant or continuous piecewise-linear approximations of the multipliers. For the purpose of stabilization we use the jumps in derivatives of the multipliers or, for piecewise constants, the jump in the multipliers themselves, across element borders. The ideas are illustrated using Poisson's equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

  • 23.
    Burman, Erik
    et al.
    Ecole Polytechnique Federale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem2005In: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 21, no 5, p. 986-997Article in journal (Refereed)
    Abstract [en]

    We stabilize the nonconforming Crouzeix-Raviart element for the Darcy-Stokes problem with terms motivated by a discontinuous Galerkin approach. Convergence of the method is shown, also in the limit of vanishing viscosity. Finally, some numerical examples verifying the theoretical predictions are presented.

  • 24.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Stabilized nonconforming finite element methods for data assimilation in incompressible flows2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 311, p. 1029-1050Article in journal (Refereed)
    Abstract [en]

    We consider a stabilized nonconforming finite element method for data assimilation in incompressible flow subject to the Stokes equations. The method uses a primal dual structure that allows for the inclusion of nonstandard data. Error estimates are obtained that are optimal compared to the conditional stability of the ill-posed data assimilation problem.

  • 25. Burman, Erik
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    The edge stabilization method for finite elements in CFD2004In: Numerical mathematics and advanced applications / [ed] Feistauer, M; Dolejsi, V; Najzar, K, BERLIN: SPRINGER-VERLAG BERLIN , 2004, p. 196-203Conference paper (Other academic)
    Abstract [en]

    We give a brief overview of our recent work on the edge stabilization method for flow problems. The application examples are convection-diffusion, with small diffusion parameter, and a generalized Stokes model.

  • 26.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    A cut finite element method with boundary value correction2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 310, p. 633-657Article in journal (Refereed)
    Abstract [en]

    In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomée in [Math. Comp. 26 (1972), 869-879]. The cut finite element method is a fictitious domain method with Nitsche-type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.

  • 27.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    A simple approach for finite element simulation of reinforced plates2018In: Finite elements in analysis and design (Print), ISSN 0168-874X, E-ISSN 1872-6925, Vol. 142, p. 51-60Article in journal (Refereed)
    Abstract [en]

    We present a new approach for adding Bernoulli beam reinforcements to Kirchhoff plates. The plate is discretised using a continuous/discontinuous finite element method based on standard continuous piecewise polynomial finite element spaces. The beams are discretised by the CutFEM technique of letting the basis functions of the plate represent also the beams which are allowed to pass through the plate elements. This allows for a fast and easy way of assessing where the plate should be supported, for instance, in an optimization loop.

  • 28.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    A simple finite element method for elliptic bulk problems with embedded surfaces2019In: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499, Vol. 23, no 1, p. 189-199Article in journal (Refereed)
    Abstract [en]

    In this paper, we develop a simple finite element method for simulation of embedded layers of high permeability in a matrix of lower permeability using a basic model of Darcy flow in embedded cracks. The cracks are allowed to cut through the mesh in arbitrary fashion and we take the flow in the crack into account by superposition. The fact that we use continuous elements leads to suboptimal convergence due to the loss of regularity across the crack. We therefore refine the mesh in the vicinity of the crack in order to recover optimal order convergence in terms of the global mesh parameter. The proper degree of refinement is determined based on an a priori error estimate and can thus be performed before the actual finite element computation is started. Numerical examples showing this effect and confirming the theoretical results are provided. The approach is easy to implement and beneficial for rapid assessment of the effect of crack orientation and may for example be used in an optimization loop. 

  • 29.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University.
    A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator2015In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 285, p. 188-207Article in journal (Refereed)
    Abstract [en]

    We consider solving the Laplace–Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.

    The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method. 

  • 30.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Augmented Lagrangian and Galerkin least-squares methods for membrane contact2018In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 114, no 11, p. 1179-1191Article in journal (Refereed)
    Abstract [en]

    In this paper, we propose a stabilized finite element method for the numerical solution of contact between a small deformation elastic membrane and a rigid obstacle. We limit ourselves to friction-free contact, but the formulation is readily extendable to more complex situations. 

  • 31.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Augmented Lagrangian finite element methods for contact problems2019In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 53, no 1, p. 173-195Article in journal (Refereed)
    Abstract [en]

    We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution. 

  • 32.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Dirichlet boundary value correction using Lagrange multipliers2019In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125Article in journal (Refereed)
    Abstract [en]

    We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to a Taylor series expansion approach previously used in the context of Nitsche’s method and, in the case of inf-sup stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary. 

  • 33.
    Burman, Erik
    et al.
    University College London, London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G
    Umeå University, Umeå, Sweden.
    Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization2018In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, no 3, article id 035004Article in journal (Refereed)
    Abstract [en]

    Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely illposed problems developed in the setting of convection-dominated convection– diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson’s equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.

  • 34. Burman, Erik
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Institutionen för matematik och matematisk statistik, Umeå universitet.
    The penalty-free Nitsche Method and nonconforming finite elements for the Signorini problem2017In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, no 6, p. 2523-2539Article in journal (Refereed)
    Abstract [en]

    We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 (2013), pp. 1295--1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.

  • 35.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Cut finite elements for convection in fractured domains2019In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 179, p. 728-736Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain, which is a union of manifolds of different dimensions such that a d dimensional component always resides on the boundary of a d+1 dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem is formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is posed on a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.

  • 36.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Finite element approximation of the Laplace–Beltrami operator on a surface with boundary2019In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 141-172Article in journal (Refereed)
    Abstract [en]

    We develop a finite element method for the Laplace–Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche’s method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order (Formula presented.) in the energy and (Formula presented.) norms that take the approximation of the surface and the boundary into account. 

  • 37.
    Burman, Erik
    et al.
    University College London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Umeå University, Sweden.
    Massing, André
    Umeå University, Sweden.
    A cut discontinuous Galerkin method for the Laplace–Beltrami operator2017In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, no 1, p. 138-169Article in journal (Refereed)
    Abstract [en]

    We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in R. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in R. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then, a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighbouring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.

  • 38.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions2019In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, no 6, p. 2247-2282Article in journal (Refereed)
    Abstract [en]

    We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in Rd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R3.

  • 39.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Zahedi, Sara
    Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.
    A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 358, article id 112645Article in journal (Refereed)
    Abstract [en]

    We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included. 

  • 40.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Zahedi, Sara
    Department of Mathematics, KTH, Stockholm, Sweden.
    Full gradient stabilized cut finite element methods for surface partial differential equations2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 278-296Article in journal (Refereed)
    Abstract [en]

    We propose and analyze a new stabilized cut finite element method for the Laplace–Beltrami operator on a closed surface. The new stabilization term provides control of the full R 3 gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace–Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and L 2 norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.

  • 41.
    Burman, Erik
    et al.
    Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Samvin, David
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    A cut finite element method for elliptic bulk problems with embedded surfaces2019In: GEM - International Journal on Geomathematics, ISSN 1869-2672, E-ISSN 1869-2680, Vol. 10, no 1, article id 10Article in journal (Refereed)
    Abstract [en]

    We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractures. In particular the Laplace–Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples. 

  • 42.
    Burman, Erik
    et al.
    Department of Mathematics, University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University.
    Stenberg, Rolf
    Department of Mathematics and Systems Analysis, Aalto University.
    Galerkin least squares finite element method for the obstacle problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 313, p. 362-374Article in journal (Refereed)
    Abstract [en]

    We construct a consistent multiplier free method for the finite element solution of the obstacle problem. The method is based on an augmented Lagrangian formulation in which we eliminate the multiplier by use of its definition in a discrete setting. We prove existence and uniqueness of discrete solutions and optimal order a priori error estimates for smooth exact solutions. Using a saturation assumption we also prove an a posteriori error estimate. Numerical examples show the performance of the method and of an adaptive algorithm for the control of the discretization error.

  • 43.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University.
    Zahedi, Sara
    KTH Royal Institute of Technology.
    Cut finite element methods for coupled bulk–surface problems2016In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, no 2, p. 203-231Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and L2 norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

  • 44.
    Burman, Erik
    et al.
    UCL, Department of Mathematics, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.
    Zahedi, Sara
    The Royal Institute of Technology (KTH), Department of Mathematics, Stockholm, Sweden.
    Stabilized CutFEM for the convection problem on surfaces2019In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 103-139Article in journal (Refereed)
    Abstract [en]

    We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove h3 / 2 order convergence in the natural norm associated with the method and that the full gradient enjoys h3 / 4 order of convergence in L2. We also show that the condition number of the stiffness matrix is bounded by h- 2. Finally, our results are verified by numerical examples. 

  • 45.
    Burman, Erik N.
    et al.
    UCL, Department of Mathematics, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.
    A cut finite element method with boundary value correction for the incompressible Stokes equations2019In: Numerical mathematics and advanced applications ENUMATH 2017, Cham: Springer, 2019, Vol. 126, p. 183-192Conference paper (Refereed)
    Abstract [en]

    We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018). 

  • 46.
    Cenanovic, Mirza
    et al.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G,
    Umeå University.
    Minimal surface computation using a finite element method on an embedded surface2015In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, no 7, p. 502-512Article in journal (Refereed)
    Abstract [en]

    We suggest a finite element method for finding minimal surfaces based on computing a discrete Laplace–Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is carried out on the piecewise planar isosurface using the shape functions from the background three-dimensional mesh used to represent the distance function. A recently suggested stabilized scheme for finite element approximation of the mean curvature vector is a crucial component of the method.

  • 47.
    Cenanovic, Mirza
    et al.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larsson, Mats G.
    Umeå University.
    Cut finite element modeling of linear membranes2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 98-111Article in journal (Refereed)
    Abstract [en]

    We construct a cut finite element method for the membrane elasticity problem on an embedded mesh using tangential differential calculus, i.e., with the equilibrium equations pointwise projected onto the tangent plane of the surface to create a pointwise planar problem in the tangential direction. Both free membranes and membranes coupled to 3D elasticity are considered. The discretization of the membrane comes from a Galerkin method using the restriction of 3D basis functions (linear or trilinear) to the surface representing the membrane. In the case of coupling to 3D elasticity, we view the membrane as giving additional stiffness contributions to the standard stiffness matrix resulting from the discretization of the three-dimensional continuum.

  • 48.
    Cenanovic, Mirza
    et al.
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larsson, Mats G.
    Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinementManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order L2-convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using L2-projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare or results to previously proposed approaches.

  • 49. Chernov, Alexey
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    An hp-Nitsche’s Method for Interface Problems with Nonconforming Unstructured Finite Element Meshes2011In: Spectral and High Order Methods for Partial Differential Equations / [ed] Jan S. Hesthaven and Einar M. Rønquist, 2011, p. 153-161Conference paper (Refereed)
    Abstract [en]

    In this paper we propose an hp-Nitsche’s method for the finite element solution of interface elliptic problems using non-matched unstructured meshes of triangles and parallelograms in  2 and tetrahedra and parallelepipeds in  3. We obtain an explicit lower bound for the penalty weighting function in terms of the local inverse inequality constant. We prove a priori error estimates which are explicit in the mesh size h and in the polynomial degree p. The error bound is optimal in h and suboptimal in polynomial degree by p 1∕2.

  • 50. Eriksson, Kenneth
    et al.
    Estep, Don
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Johnson, Claes
    Introduction to Adaptive Methods for Differential Equations1995In: Acta Numerica, ISSN 0962-4929, E-ISSN 1474-0508, Vol. 4, p. 105-158Article in journal (Refereed)
    Abstract [en]

    Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719).

    When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results…. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used…. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).

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  • rtf