Local-Mesh, Local-Order, Adaptive Finite Element Methods with a Posteriori Error Estimators for Elliptic Partial Differential Equations.

摘要

The traditional error estimates for the finite element solution of elliptic partial differential equations are a priori, and little information is available from them about the actual error in a specific approximation to the solution. In recent years, locally-computable a posteriori error estimators have been developed, which apply to the actual errors committed by the finite element method for a given discretization. These estimators lead to algorithms in which the computer itself adaptively decides how and when to generate discretizations. So far, for two-dimensional problems, the computer-generated discretizations have tended to use either local mesh refinement, or local order refinement, but not both. In this thesis, we present a new class of local-mesh, local-order, square finite elements which can easily accommodate computer-chosen discretizations. We present several new locally-computable a posteriori error estimators which, under reasonable assumptions, asymptotically yield upper bounds on the actual errors committed, and algorithms in which the computer uses the error estimators to adaptively produce sequences of local-mesh, local-order discretizations.