`A posteriori´ element by element local error estimation technique and 2D & 3D adaptive finite element mesh refinement

A new approach for estimating an `a posteriori´ error locally on an element by element basis for the adaptive refinement of a class of 2D and 3D boundary value problems has been investigated in this paper. It is necessary to have an efficient, robust and reliable error estimator to generate an optimal adaptive mesh. In this work an `a posteriori´ error is computed by solving a local problem on a patch of elements. It is simple to implement and is computationally inexpensive. This method computes the local as well as global error by using a h-version of adaption with quadratic shape functions to solve the local problem. The refinement algorithm makes use of a minimal hierarchical tree based data structure which minimizes the amount of tree traversal normally required during the refinement process. The efficiency of the local error estimation technique has been demonstrated by the adaptive refinement of an ac transmission line problem in 2D and an eddy current problem employing complex magnetic vector potential formulation in 3D. The coarse mesh and the refined optimal meshes and the numerical results substantiate that the local `a posteriori´ error estimate is efficient and simple to use for most practical applications