An Adaptive Dual Reciprocity Scheme for the Numerical Solution of Poisson's Equation 

Juan Jose Rodriguez from the Central University of Venezuela in Caraca has successfully passed his PhD viva at WIT with a thesis on "An Adaptive Dual Reciprocity Scheme for the Numerical Solution of Poisson's Equation".

Juan Jose is a Computer Scientist by training and a Professor at the School of Architecture of the Central University. Although his background was different from most of the students undertaking boundary element research at WIT, he was able to complete an excellent and original contribution, and he received the congratulations of the examiners, particularly his external Professor, Bill Hall, from Teeside University.

Juan Jose presented a novel mesh refinement strategy and adaptive technique for the numerical solution of the Laplace and Poisson equations using the Boundary Element Method (BEM) and the Dual Reciprocity Method (DRM), respectively. Numerical and computational techniques were derived from error analyses. Error indicators are based on a collocation scheme using uniform norms that compare values of field variables at two successive mesh iterations. For problems governed by the Laplace equation the adaptation is achieved through mesh refinements of type h, ie subdividing selected elements of one iteration to produce the next. In brief, when a new mesh I is constructed and solved during the adaptive process, the field variable approximations obtained for all new collocation nodes are compared with the values calculated at the same coordinates using the solution of the previous smaller mesh i-1. The comparison is performed using an ad hoc indicator derived from error bounds of the piecewise polynominal collocation approximation of Fredholm integral equations. The results obtained with this technique indicate that it provides a good foundation to appraise the discretization error of the Boundary Element Method.

For problems governed by the Poisson equation a similar technique is developed for the numerical solution obtained from the Dual Reciprocity Method. The internal domain is divided into non-overlapping clusters containing a number of Dual Reciprocity (DR) points. Error indicators are evaluated for each cluster to determine whether or not an increase of DR points is needed. As for the Laplace problems, uniform norms are used to compare values of the potentials at the DR points from two consecutive iterations. The proposed analyses and the case studies show that the numerical error is reduced when the boundary elements are refined simultaneously with the DR internal points.

The thesis represents a valuable contribution for the better understanding of the Dual Reciprocity Method, which was originally devised by the Director of WIT, Professor Carlos Brebbia in 1982 in collaboration with one of his researchers.