Second-order elliptic PDEs with discontinuous boundary data

Abstract

We consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well defined in the standard H 1 −H 1 setting we introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we discuss the numerical solution of such problems. Specifically, we employ an hp-discontinuous Galerkin method and derive (enhanced) L 2 -norm upper and local lower a posteriori error bounds. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h-and the hp-version setting. Indeed, in the latter case, exponential convergence of the error is attained as the mesh is adaptively refined.

Discontinuous Galerkin methods for problems with Dirac delta source

“…Here, in the context of conforming FEM, we mention the a priori results in [7,17], as well as the a posteriori error analysis in [3]. For DG approximations to low-regularity problems, see, e.g., [13,19].…”

“…Pursuing a similar approach as in [13,14] (i.e., by choosing a suitable smooth nonnegative cut-off function on ω e and by applying norm equivalences in finite dimensional spaces), we find an auxiliary function …”

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

“…The norms of these spaces contain local radial weights at the discontinuity points A of the Dirichlet boundary data, and, thereby, account for possible singularities in the solution of (1)- (2). Based on an inf-sup theory, the work [7] shows that (1)-(2) exhibits a unique solution within this framework.…”

“…for any v ∈ H 2 (Ω) ∩ H 1 0 (Ω), where we write n for the unit outward normal vector to the boundary Γ. Alternatively, the following saddle point formulation, which traces back to the work [9], may be applied: provided that g ∈ H 1 /2−ε (∂Ω), for some ε ∈ [0, 1 /2), find u ∈ H 1−ε (Ω) with u| Γ = g such that for all v ∈ H 1+ε (Ω) ∩ H 1 0 (Ω); for results dealing with finite element approximations of (3), we refer to [4]. Another related approach is based on weighted Sobolev spaces (accounting for the local singularities of solutions with discontinuous boundary data), and has been analyzed in the context of hp-type discontinuous Galerkin methods in [7].…”

“…for all v ∈ H 1+ε (Ω) ∩ H 1 0 (Ω); for results dealing with finite element approximations of (3), we refer to [4]. Another related approach is based on weighted Sobolev spaces (accounting for the local singularities of solutions with discontinuous boundary data), and has been analyzed in the context of hp-type discontinuous Galerkin methods in [7].…”