Boundary Element Method Applied to the Solution of Partial Differential Equations

CONCLUSIONSIn the present thesis, the boundary element method is presented in details. The boundary integral equations for Laplace and Poisson equations are derived.From the present study the main points are obtained:1- An error analysis concerning the accuracy of the boundaryElement method show that;a) The best accuracy is obtained for the Gauss integration points equal eight.b) The accuracy in solution is improved using quadratic boundary elements2. An error analysis concerning the reduction of domain integral show that the cell method gives better resultswhen the domain (1x1) is subdivided into 16 cells (table 3.9)3. We did apply the boundary element method for the equation for the evaluation of the temperature distribution inside three triangular domain subjected to Laplace equation and found in each of the applications that the error in calculations using boundary elements was found smaller than using finite elements for equal mesh size.4. In problem 1 we did first plot the results in fig.3.10 for linear boundary elements, and finite elements.5. We then tested, the quadratic boundary elements for the same problem. The results obtained where plotted in fig.3.11, showing an improvement in accuracy.6. We did plot the error in boundary and finite element, for two horizontal sections in the triangle.7. A similar approach was followed, for the solution of problems 2 and 3. Linear boundary elements where used together with the finite element for an equal mesh size. The results where plotted in fig(3.15), (3.19) and (3.20). The error in evaluating temperature nearly drop to zero using quadratic boundary elements as it appears in Fig (3.19), (3.20)8. Poisson equation was solved using boundary element method and the method of cells. The results obtained where compared to the exact solution whenever possible.9. The results obtained in chapter 3 show that the boundary element method accuracy is higher than the finite element method, for equal mesh size and similar linear interpolating functions.