Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Abstract

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.

Keywords:

• boundary element method
• collocation
• Laplace equation
• Richardson extrapolation

2000 AMS Subject Classifications :

• Primary 65B05
• 65N35
• 65N38
• Secondary 35J05