
Error EstimatesTo illustrate the WEBmethod in an elementary setting, we consider a simple model problem, which has an analytic solution and, thus, permits the computation of exact approximation errors. The simulation domain, shown above, is bounded by an ellipse and a circle. Numerical solutions are computed for Poisson's equation with homogeneous Dirichlet boundary conditions. The righthand side is chosen from a given exact solution, which is depicted below. For the construction of the WEBbasis we used the analytic weight function The error was measured in the following two norms. For Bsplines of order 2 to 6 (polynomial degree 1 to 5) the grid width was succesively halfed and the RitzGalerkin approximations computed. In the following diagram the L2error was plotted versus the number of basis functions. Dividing consecutive errors for a fixed order and taking logarithms with respect to the base 2 yields estimates for the convergence rate. The diagram below confirms that the optimal rates are attained. The H1norm corresponds to the natural energy norm associated with Poisson's operator. The optimal approximation order is equal to the degree of the WEBsplines (order minus 1). Again, the numerical results correspond to the theoretical predictions. The qualitative behavior of the error is similar as for the L2norm. The following diagram compares the L2error for WEBsplines to that of linear trial functions on a triangulation (black caro markers). Because of the stabilization of the Bsplines, the RitzGalerkin system can be solved efficiently with standard iterative methods, such as the preconditioned conjugate gradient algorithm. The following diagram shows the number of pcgiterations for a tolerance of 1e14 as a function of the dimension of the system. LiteratureK. Höllig, U. Reif, J. Wipper: Weighted extended bspline approximation of Dirichlet problems. SIAM Journal on Numerical Analysis, Volume 39, Number 2, pp. 442462, 2001.See Publications for a complete list of WEBpublications.
Author: Joachim Wipper ; Last modification: 2006/08/31 09:36:15 UTC.
