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

Abstract

Abstract. We present a numerical approximation method for linear diffusionreaction 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 H 2 -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 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 with respect to the mesh size.

An Eclectic View on Numerical Methods for PDEs: Presentation of the Special Issue “Advanced Numerical Methods: Recent Developments, Analysis and Applications”

“…The first comprises two articles that present general analysis frameworks for parabolic and hyperbolic models, by Gallouët [14] and Abgrall [1], respectively; the settings and results presented in these papers apply to and help create links between various numerical methods. In the second group, linear elliptic equations are considered with data or solutions that lack full regularity; boundary conditions are weakly imposed using Nitsche's technique and convergence results are obtained, using semi-analytical techniques by Baumann and Wihler [4] or generalisations of Strang's lemmas with weak notions of traces and mollifying tools by Ern and Guermond [12]. The preservation of physical or asymptotic properties is concerned by four contributions that make up the third group; Cancès, Chainais-Hillairet and Krell in [7] design a non-linear scheme to preserve, on generic meshes, the expected energy dissipation relation of a convection-diffusion model; Brenner, Çeşmelioǧlu, Cui and Sung in [6] focus on the questions of eliminating spurious modes in a fluid-structure model; the issue of loss of accuracy due to poor mass conservation in the approximation of incompressible Navier-Stokes equations is covered by Ahmed, Linke and Merdon in [2]; finally, the design of well-balanced and asymptotic preserving schemes for hyperbolic systems, using local exact solutions, is the topic of the paper by Morel, Buet and Després [17].…”