A posteriori error analysis for finite element solution of one-dimensional elliptic differential equations using equidistributing meshes

Abstract

The paper is concerned with the adaptive linear finite element solution of linear one-dimensional elliptic differential equations using equidistributing meshes. A strategy is developed for defining such meshes based on a residual-based a posteriori error estimate. The mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation (i.e., the mesh equation). An iterative algorithm is proposed for solving this system for the mesh and the finite element solution. The existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh, which is also established in the current work. Error bounds for the finite element solution are obtained for the equidistributing and quasi-equidistributing meshes. They show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes. The departure from the equidistributing meshes has only a mild effect on the finite element error. Numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings.