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An Accurate High-Order Method to Solve the Helmholtz Boundary Value Problem for the 3D Laplace Equation


Abstract

The 3D Laplace equation is one of the important PDEs of physics and describes the phenomonology of electrostatics and magnetostatics. Frequently very precise solution of this PDE is required; but with conventional finite element or finite difference codes this is difficult to achieve because of the need for an exceedingly fine mesh which leads to often prohibitive CPU time.

We present an alternate approach based on high-order quadrature and a high-order finite element method. Both of the ingredients become possible via the use of high-order differential algebraic methods. Various examples of the method and the precision that can be achieved will be given. For example, using only about 100 finite elements of order 7, accuracies in the range of 10-6 can be obtained in the 3D case.


S. Manikonda, M. Berz, International Journal of Pure and Applied Mathematics 23,3 (2005) 365-378


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