Maximal Lp-Lq regularity for discrete elliptic differential operators

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Maximal regularity estimates play an important role in the analysis of parabolic problems. In particular, maximal regularity estimates are very useful for nonlinear parabolic problems. Therefore, in this book, maximal regularity estimates for approximations of solutions of parabolic differential equations by finite elements or finite differences are investigated. In this context, one has to show that maximal regularity estimates are independent of the mesh. More precisely, uniform estimates for a family of finite element (finite difference) operators are shown. These estimates are used to prove uniqueness and existence of a local, strong solution for approximations of semilinear problems. Moreover, uniform maximal regularity estimates lead to a new approach for proving error estimates. Finally, error estimates in the linear and semilinear case are proved.