CLOSED-FORM EVALUATION OF FLUX INTEGRALS APPEARING IN A FEM SOLUTION OF THE 2D POISSON EQUATION WITH DIPOLE SOURCES

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 dip ole sources. For this important class of problems, an accurate solutio n may be obtained by using a subtraction formulation, in which the unk nown is the original potential function minus the potential of the dip ole in infinite homogeneous space. This formulation requires the evalu ation of certain flux integrals that appear at the boundaries of the e lements. Computing these flux integrals numerically requires considera ble 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 err ors in the solution for the potential, since the FEM matrix may often be ill-conditioned. A closed-form evaluation of these flux integrals f or first- and second-order triangular elements along linear or quadrat ic edges is presented here. It is shown that these closed-form express ions reduce the computation time spent in the construction of the righ t-hand side vector by more than a, factor of 5 compared to adaptive Ga ussian quadrature, and eliminate the large errors that are due to ill- conditioning.