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A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions

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Published/Copyright: December 13, 2017
Computational Methods in Applied Mathematics
From the journal Computational Methods in Applied Mathematics Volume 18 Issue 3

Abstract

We present a numerical approximation method for linear elliptic diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known singular functions as well as of an H2-regular part. The latter part is expressed in terms of an elliptic problem with regularized Dirichlet boundary conditions, and can be approximated by means of a Nitsche finite element approach. The discrete solution of the original problem is then defined by adding back the singular part of the exact solution to the Nitsche approximation. In this way, the discrete solution can be shown to converge of second order in the L2-norm with respect to the mesh size.

Keywords: Second-Order Elliptic PDE; Discontinuous Dirichlet Boundary Conditions; Nitsche FEM
MSC 2010: 65N30

References

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Received: 2017-06-10
Revised: 2017-11-14
Accepted: 2017-11-19
Published Online: 2017-12-13
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. An Eclectic View on Numerical Methods for PDEs: Presentation of the Special Issue “Advanced Numerical Methods: Recent Developments, Analysis and Applications”
  3. Some Remarks About Conservation for Residual Distribution Schemes
  4. Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations
  5. A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions
  6. A Nonconforming Finite Element Method for an Acoustic Fluid-Structure Interaction Problem
  7. Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations
  8. Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation
  9. Abstract Nonconforming Error Estimates and Application to Boundary Penalty Methods for Diffusion Equations and Time-Harmonic Maxwell’s Equations
  10. Discrete Functional Analysis Tools for Some Evolution Equations
  11. A Posteriori Stopping Criteria for Optimized Schwarz Domain Decomposition Algorithms in Mixed Formulations
  12. Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P1 Model
https://doi.org/10.1515/cmam-2017-0057
Keywords for this article
Second-Order Elliptic PDE; Discontinuous Dirichlet Boundary Conditions; Nitsche FEM

Articles in the same Issue

  1. Frontmatter
  2. An Eclectic View on Numerical Methods for PDEs: Presentation of the Special Issue “Advanced Numerical Methods: Recent Developments, Analysis and Applications”
  3. Some Remarks About Conservation for Residual Distribution Schemes
  4. Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations
  5. A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions
  6. A Nonconforming Finite Element Method for an Acoustic Fluid-Structure Interaction Problem
  7. Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations
  8. Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation
  9. Abstract Nonconforming Error Estimates and Application to Boundary Penalty Methods for Diffusion Equations and Time-Harmonic Maxwell’s Equations
  10. Discrete Functional Analysis Tools for Some Evolution Equations
  11. A Posteriori Stopping Criteria for Optimized Schwarz Domain Decomposition Algorithms in Mixed Formulations
  12. Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P1 Model
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