A CONTROL THEORETIC APPROACH TO MODEL REDUCTION OF PDES – MOVING MESH METHODS

Abstract

Moving mesh methods is a class of methods for model reduction of PDE models, based on a dynamically moving discretization mesh. Moving mesh methods have been widely used for solving differential equations involving large solution variations. The methods can roughly be divided into moving finite difference methods (MFD) and moving finite element methods (MFEM). In this paper we consider these methods from a feedback control point of view and use results from control theory to provide a plausible explanation for the robustness problems encountered in most of the methods. Based on these results we also propose a novel moving finite element method, OCMFE, in which the error introduced by the spatial discretization is estimated based on residual calculations and a simple feedback control algorithm is employed to adjust the size of the various elements such that the estimated model reduction error is equidistributed over the spatial domain.