Uniform Pointwise Convergence of an Upwind Finite Volume Method on Layer-Adapted Meshes

Abstract

A singularly perturbed convection‐diffusion problem is considered. The problem is discretized using an inverse‐monotone finite volume method on Shishkin meshes. We establish first‐order global pointwise convergence no matter how small the perturbation parameter. We show both theoretically and experimentally that certain choices of the stabilization parameter are particularly favourable. Numerical experiments support the theoretical results.

A Numerical Method for Singularly Perturbed Convection-diffusion Problems Posed on Smooth Domains

“…Remark 3. Linß [11] examined a class of fitted finite difference operators (arising from a finite volume formulation) on a tensor product of Shishkin meshes. Using the (L ∞ , L 1 ) stability argument developed by Andreev [1], Linß established that for u ∈ C 4 ( Ω), then…”

“…248]). If we formally set (11), the resulting finite difference scheme scheme fits into the framework of fitted finite difference schemes analysed in Linß [11].…”

A New Upwind Finite Volume Element Method for Convection-Diffusion-Reaction Problems on Quadrilateral Meshes

“…FVEM has not only low-order element schemes [8][9][10][11][12][13][14], but also high-order element schemes [15][16][17][18][19][20][21][22]. Many first-order finite volume methods are constructed using upwind technique for convection-diffusion equations [7,[23][24][25][26][27][28][29][30].…”

Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method

“…The numerical simulation and the error estimate of steady-state convection-dominated diffusion problem was investigated in [31] using a cell-vertex finite volume approximation, whereas [32] used cell-centered finite difference approximations for a steady-state convection-diffusion problem to get second-order convergence that fulfilled the discrete maximum principle with discrete Sobolev spaces. By looking at the work in [33] a singular perturbed convection-diffusion problem was discretized with the help of the upwind finite volume scheme with uniform point-wise convergence over layer-adapted meshes. The authors [34] discussed the numerical solution of the unsteady-state advection-diffusion problem in two dimensions using the upwind finite volume method and an alternating-implicit, upwind finite volume method.…”