Analysis of Discontinuous Galerkin Methods using Mesh-Dependent Norms and Applications to Problems with Rough Data
Provides theoretical error bounds for discontinuous Galerkin methods applied to elliptic problems with rough data, addressing a gap in numerical analysis for non-standard solution regularity.
The paper proves inf-sup stability of a discontinuous Galerkin scheme in mesh-dependent norms, yielding a priori error bounds, and demonstrates quasi-optimal error control for problems with rough source terms where weak solutions are not available.
We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine a problem with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control.