Finite Differences and Volumes
In this chapter, we describe the finite-difference and finite-volume discretization methods for scalar second-order elliptic PDEs such as the Poisson equation, highly anisotropic diffusion equations, the indefinite Helmholtz equation, and the convection diffusion equation in two spatial dimensions. We study not only the accuracy of the discretization method (the rate by which the discretization error approaches zero together with the meshsize) but also its adequacy (the rate by which the discretization error approaches zero together with the meshsize and the parameter used in the PDE). Finally, we present the finite-volume discretization method for diffusion problems with variable or even discontinuous coefficients.
KeywordsDirichlet Boundary Condition Poisson Equation Truncation Error Helmholtz Equation Discretization Method
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