Extrapolation techniques for reducing the pollution effect of reentrant corners in the finite element method (Q1106002)

Finite element solutions of boundary value problems with reentrant corners suffer from global loss of accuracy caused by the influence of singularities. A sufficiently simple extrapolation technique is suggested in order to subpress this effect, for application of which an a priori knowledge of the asymptotic growth of the singularities is required. One of the main results is an asymptotic representation of the error \[ (1)\quad (u-u_ n)(x)=\sum^{N}_{n=1}A_ nK_{n;1}(u)S'_{n;- 1}(x)h_ n^{2\alpha_{n;1}}+O(h^ 2| \log h|), \] where N is the number of reentrant corners, \(h_ n\) are the local mesh size parameters, \(K_{n;1}\) and \(S'_{n;-1}\) are the corresponding leading stress intensity factors and ``dual'' singular functions. This representation holds on certain locally uniform meshes. On the basis of (1) an extrapolation procedure is proposed for eliminating the N dominant error terms yielding approximations \(\tilde u_ n\) and \(K_{n;1}(\tilde u_ n)\) which are of almost optimal order \((u-\tilde u_ n)(x)=O(h^ 2| \log h|),\quad K_{n;1}(u)-K_{n;1}(\tilde u_ n)=O(h^ 2).\) The procedure can be directly combined with existing routines. It can be applied to higher order elements and to other boundary value problems. Numerical experiments support the theoretical results.