Second-order elliptic PDEs with discontinuous boundary data

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 H¹ – H¹ 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) L²-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.