On an Adaptive Grid Refining Technique for Finite Element Approximations

Abstract

We consider a family of finite element spaces and minimize an energy functional over each space. The space which allows the lowest energy is considered “optimal.” Such a family is constructed by starting with an initial “triangulation” and refining one or more “triangles” at a time. We estimate the profit in energy gained by refining a triangle and set up a discrete optimization problem which determines the optimal refinement strategy according to a prescribed bound of the costs. This enables us to construct the final grid by using as few as possible intermediate grids. Instead of solving the original optimization problem we set up a partially dualized form of it which produces a nearly optimal solution and can be solved very efficiently.