The present article is the second one of a trilogy of papers [see the foregoing and the following entry (Zbl 0723.73074; Zbl 0723.73076)] on the development of a adaptive h-p version of the finite element method for the solution of linear elliptic boundary-value problems (BVPs) characterized by general elliptic systems of partial differential equations.
In Part 2 of the presentation, the authors address the question of a posteriori error estimation of the h-p finite element schemes and furthermore analyze and experiment with five methods for elliptic BVPs. These techniques are referred to as the residual estimation method, the duality method, the subdomain-residual method, a method based on interpolation theories, and a post-processing method. In particular, element residual methods (1) are based on the computation of the residual over each element using the data in special elementwise BVPs for the local error \(e_ h\). Duality methods (2) are valid for self-adjoint elliptic problems using the duality theory of convex optimization to derive upper and lower bounds of the element errors. Subdomain-residual methods (3) formulate the local error in a given element over a patch of elements surrounding the element. Interpolation methods (4) use the interpolation theory of finite elements in Sobolev norms to produce rapid estimates of the local error over individual elements and, finally, post- processing methods (5) estimate the error by comparing a post-processed version of the approximate solution with the approximate solution itself.
The different methods of residual estimation are designed to be used repeatedly during the evolution of a changing h-p mesh. A lot of numerical examples illustrate the efficiency of the methods.