Closed-Form Evaluation of Flux Integrals Appearing in a Fem Solution of the 2D Poisson Equation with Dipole Sources

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Abstract
The finite element method (FEM) is a versatile method for numerically solving the 2-D Poisson equation with arbitrary inhomogeneity. In many applications, the sources that appear in the Poisson equation are dipole sources. For this important class of problems, an accurate solution may be obtained by using a subtraction formulation, in which the unknown is the original potential function minus the potential of the dipole in infinite homogeneous space. This formulation requires the evaluation of certain flux integrals that appear at the boundaries of the elements. Computing these flux integrals numerically requires considerable computation time, especially for element edges that are very close to the dipole, because the field is rapidly varying near the dipole. Furthermore, numerical errors in the computation may produce large errors in the solution for the potential, since the FEM matrix may often be ill-conditioned. A closed-form evaluation of these flux integrals for first- and second-order triangular elements along linear or quadratic edges is presented here. It is shown that these closed-form expressions reduce the computation time spent in the construction of the right-hand side vector by more than a factor of 5 compared to adaptive Gaussian quadrature, and eliminate the large errors that are due to ill-conditioning.

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