Regularity and error estimators for elliptic problems with discontinuous coefficients
Martin Petzoldt · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2001
We regard linear elliptic equations with a discontinuous diffusion coefficient k in two and three space dimensions. The coefficient k is constant on polygonal (polyhedral) subdomains. These problems are also known as Laplace interface problems. It is known that solutions of these problems have lower regularity due to singularities. In the second chapter we derive Sobolev H (s)-regularity, where s belongs to (1,2) which hold independently of the shape of the subdomains. We use a known criterion on the structure of coefficients \- the quasi-monotonicity condition - to give regularity results in Sobolev spaces H (1+1/4) independent of the jump size of the coefficients. We argue that the quasi-monotonicity is also a necessary condition for higher regularity independent of the jump size of k. Further we give sharp regularity results which depend on the jump size. We show that a checkerboard like distribution of values for the coefficient k leads to the worst possible regularity. For the regularity results in 3D we use the derived 2D results. In the third chapter of this thesis we discretize the problem with linear finite elements. We propose treatment of the arising singularities by a posteriori mesh refinement on the basis of new a posteriori error estimators. If the quasi-monotonicity condition is fulfilled, we show that the a posteriori error estimators bound the discretization error from above with constants which do not depend on the jump size of the coefficient. For a lower bound of the error the quasi-monotonicity condition is not needed. In various numerical examples (chapter 4) we confirm the applicability of the derived error estimators to problems with singularities. The examples comprise model problems, problems with real data from groundwater flow and 3D examples. The examples show that mesh refinement lead to error reduction rates in terms of unknowns N to the power of (-1/space dimension), which are expected to be optimal. The ratio of the error estimator and the true error takes on problem independent moderate values.