Adaptive finite element methods in computational mechanics

In this paper, we review the general approach to adaptivity for finite element methods presented in [l-16]. We also present new theoretical and computational results for linear elasticity, non-linear elasto-plasticity and non-linear conservation laws illustrating the general theory. The basic problem in adaptivity for finite element methods may be formulated as follows. Suppose 9 is a given (initial-)boundary value problem with given data f and corresponding exact solution U. Let ??,, p be a finite element method for 9 based on piecewise polynominal approximation on a family { .Yh,p} of meshes with mesh functions h = h(x, t) and p = p(x, t) giving the local mesh size h(x, t) in space and time and local order p(x, t) of the polynomial approximations as functions of space x and time t and let { uh,*} be the corresponding family of finite element solutions. Let I] * 11 b e a g iven norm and TOL > 0 a given tolerance. Then construct and algorithm & for finding a mesh Y,_, such that for the corresponding finite element solution u,,~, we have