Numerical construction of optimal adaptive grids in two spatial dimensions
Disregarding work on optimal grids for the numerical solution of boundary value problems done by Bakhvalov, and also disregarding a large number of more recent papers on grids for two-dimensional convection-diffusion problems, see e.g. papers by Shishkin, Roos, Stynes, the authors consider a practical algorithm to obtain triangular grids for usual finite elements in the case of convection dominated problems. For this they start from a coarse uniform grid, compute the finite element solution and from here a partition of points along a special flow line, and then construct further points on level curves orthogonal to the flow line. Then follows a Delaunay triangulation of the obtained grid points. Many figures are shown with grids resulting for three academic examples (with constant velocity vector in the convection-diffusion equation). Here, it is visible that very long thin triangles may result, but the algorithm seems capable to catch a curved inner layer. Optimality of the grids is claimed since the \(H^1\)-norm of the error drops linearly with \(n^{-1/2}\), where \(n\) is the number of triangles.
scientific article; zbMATH DE number 1471214
scientific article