Adaptive Recovery of near Optimal Meshes in the Finite Element Method for Parameter Dependent Problems

A method for adaptive construction of nearly optimal nite element meshes is presented. The method is based on the notion of mesh density. For parametrized problems optimal meshes for new parameter values can be predicted from previous meshes. The method is of the one shot type, normally giving a suuciently good approiximation to the optimal mesh in the rst attempt , but when the rst shot is unsuccesful the method can be iterated. For parametrized problems no \coarse mesh" solutions are required except for the very rst problem and whenever the method fails in providing an acceptable mesh, and iteration becomes necessary. 1. Introduction The idea of selecting nite element meshes not simply according to physical properties of the object that is being modelled, but to select the meshes in order to achieve a certain accuracy of the nite element solution; this idea is about as old as the nite element method itself. Normally the goal in the mesh construction process is to construct a mesh which is optimal or nearly optimal in the sense that it provides a solution within a prespeciied tolerance from the exact solution to the problem considered, and that the mesh used at the same time is as coarse or nearly as coarse as possible such that the work involved in nding the nite element solution from the mesh is as little as possible. The precise deenition of optimality varies from author to author, but the core of all approaches is as stated above. Below the deenition used in this paper will be speciied. Some approaches base the selection of meshes entirely on a priori knowledge about the problem and its solution. Those approaches are generally not as precise as methods based on a posteriori knowledge about the solution obtained after an initial solution on a coarse mesh. (That is they provide meshes further from the optimal than a posteriori methods). A posteriori methods can be divided into two main groups of iterative and one step methods. In the iterative methods the nal nearly optimal mesh is found after a series of steps starting from the initial coarse mesh and the solution on this mesh. Each step involves a reenement of the old