Features of a time-dependent fundamental solution in the Green element method

Abstract

New sets of element coefficients are derived for the Green element method (GEM) which incorporates the time-dependent fundamental solution of the linear diffusion differential operator in two spatial dimensions. These coefficients are obtained for both rectangular and triangular grids when Green element calculations are carried out for heat transfer and contaminant transport problems. Similar coefficients had earlier been derived for these two problems in one spatial dimensions [Appl. Math. Model. 22 (1998) 687; Eng. Anal. Boundary Elements 23 (1999) 577]. The flexibility offered by GEM, that allows the integral representation of the differential operator to be evaluated strictly within a typical element, is exploited by switching the order of integration in time and space to achieve, to a large extent, exact expressions of the element coefficients. In this way the accuracy of the numerical calculations is preserved. Comparison of the current formulation with an earlier one that incorporates the logarithmic fundamental solution indicates that the current formulation does better than the previous one for the heat transfer problem, but the reverse is the case for the contaminant transport problem. As with the 1-D formulation, the current 2-D formulation produces better solutions for the transport problem using larger time step––a numerical feature previously established by the numerical stability analysis [Eng. Anal. Boundary Elements 23 (1999) 577].