Galerkin Boundary Element Method For Dirichlet Problem Of 2-D Heat Conduction Equation

The heat conduction equation is a typical and simple parabolic partial differential equation with wide applications. Since Boundary Element Method (BEM) was applied to numerical computation, the BEM solution was implemented to heat conduction equation usually by direct method, while indirect method was investigated theoretically.Collocation method is a convenient and practical method to solve boundary integral equation, so it was widely used in engineering and nearly all BEM monograph with applications introduce collocation method only. The program in those monograph are all deal with collocation method. The Galerkin method based on indirect boundary integral equation was rarely implemented on numerical computation since it has trouble with quadruple integral calculation.In this paper, Dirichlet problem of two-dimensional heat conduction equation is considered. By adopting time-dependent fundamental solution, indirect boundary integral equation and its equivalent Galerkin variational formula which based on simple layer potential are derived. The method comes down to quadruple integral calculation on space-time. In our implementation, the Galerkin variational equation is discretized by constant elements and cells, the integration for time is carried out analytically, while the space integrations are carried out by combination of analytical integral and Gaussian integral. We deduced the analytical integral formulas for the calculation of singular integrals and other formulas for numerical calculation, and finished a Fortran90 programme, several numerical examples illustrate the feasibility and the efficiency of the proposed method.