A priori regularity results for discrete solutions to elliptic problems - Toni Scharle (2020)
This thesis is concerned with the development and analysis of a discrete counterpart of the well-known De-Giorgi-type regularity theory for solutions of elliptic partial differential equations in the setting of finite element approximations. We consider a finite element space consisting of piecewise affine functions on shape-regular triangulations of polyhedral Lipschitz domains Ω ⊂ R n . We identify conditions for the mesh and the data to prove a global a priori Hölder-norm bound for finite element approximations of solutions to linear elliptic equations of the form −div(A∇u) = f − divF, where A ∈ L ∞ (Ω;R n×n ) is a uniformly elliptic matrix-valued function and F ∈ L p (Ω;R n ) and f ∈ L q (Ω) are given functions for p > n and q > n/2 . After proving a Caccioppoli-type inequality for discrete subsolutions, we use iteration techniques to establish a priori L ∞ - and Hölder-norm bounds. In particular, all estimates are proved for adaptively refined, highly graded meshes.