The Performance of Numerical Methods for Elliptic Problems with Mixed Boundary Conditions

..We consider solving linear, second order, elliptic par~ial differential equations with boundary conditions of type Dirichlet (Dffi), mixed (MIX) and Neumann (NEU) and using software modules which implement five numerical methods: Hennite collocation plus band Gauss elimination (HC), ordinary finite differences plus band Gauss elimination (SP), ordinary finite differences with Dyakanov iteration (DY), DY with Richardson extrapolation to achieve fourth order convergence (D4), and ordinary finite differences with multigrid iteration (MG). We carry out a performance evaluation in which we measure the grid size and the computer time needed to achieve three significant digits of accuracy in the solution. We compute the changes in these two mell.BUl"eS as we change boundary condition types from Dffi to MIX and MIX to NEU and then test the following hypotheses: 1) the performance of all the modules is degraded by introducing the derivative terms into the boundary conditions, 2) HC is least affected, 3) the fourth order modules (HC and D4) are less affected than the other second order modules, 4) 6P is most affected. We establish these hypotheses with high levels of confidence U8ing a sample of problems. We also establish with considerable confidence that these modules have the following rankings in absolute comparative time perfonnance: MG (best), HC and D4, DY and sP (worst). ·Supported in part by Air Force Office of Scientific Research grant AFOSR 84-0385.