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The performance of numerical methods for elliptic problems with mixed boundary conditions
Author(s) -
Dyksen Wayne R.,
Ribbens Calvin J.,
Rice John R.
Publication year - 1988
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690040407
Subject(s) - mathematics , finite element method , boundary value problem , dirichlet boundary condition , collocation (remote sensing) , orthogonal collocation , neumann boundary condition , boundary (topology) , collocation method , elliptic operator , mathematical analysis , ordinary differential equation , differential equation , computer science , physics , machine learning , thermodynamics
We consider solving linear, second order, elliptic partial differential equations with boundary conditions of types Dirichlet (DIR), mixed (MIX), and nearly Neumann (Neu) by using software modules that implement five numerical methods (one finite element and four finite differences). They represent both the new generation of improved methods and the traditional ones; they are: Hermite collocation plus band Gauss elimination (HC), ordinary finite differences plus band Gauss elimination (5P), ordinary finite differences with Dyaknov 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 measures as we change boundary condition types from DIR to MIX and MIX to NEU and then test the following hypotheses: (i) the performance of all the modules is degraded by introducing the derivative terms into the boundary conditions; (ii) finite element collocation (HC) is least affected; (iii) the fourth order modules (HC and D4) are less affected than the other second order modules; and (iv) the traditional 5‐point finite differences (5P) are most affected. We establish these hypotheses with high levels of confidence by using several sample problems. The most significant conclusion is that a high order collocation method is preferred for problems with general operators and derivatives in the boundary conditions. We also establish with considerable confidence that these modules have the following rankings in absolute comparative time performance: MG (best), HC and D4, DY, and 5P (worst).