A priori regularity results for discrete solutions to elliptic problems

Thesis

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.

Next, we establish a local L ∞ -norm bound for finite element approximations of solutions to p-Laplacian systems on non-obtuse meshes. Again, we do not require our mesh to be quasi-uniform but we do allow highly graded meshes. As there is no natural notion of a positive part for vector functions, we use different techniques than in the scalar, linear uniformly elliptic case.

We then apply the results to the finite element approximation of solutions to a system that describes the steady flow of a chemically reacting incompressible fluid. The convergence of a series of approximations was already established in the literature in two space dimensions, but the lack of a De-Giorgi-type regularity theory in the finite element setting prevented a direct generalisation to the physically relevant case of three space dimensions. We show that the theory that was developed in this thesis is under certain restrictions strong enough to overcome this limitation, thus enabling us to extend the convergence theory for finite element approximations of the problem to three space dimensions.

Authors: Toni Scharle

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English