Grid-projection solution of an elliptic problem for an irregular grid

Abstract

A new grid-projection method is constructed for solving elliptic problems with a small parameter at highest derivatives on adaptive irregular grids. A direct rather than a variational problem formulation is used. The approximate solution in every cell is a quadratic polynomial. The coefficients of the polynomial are determined from the collocation (least-squares) equations and consistency conditions or boundary conditions. The conditions for linear combinations of the solution and its derivative along the normal to the boundary between the cells to be continuous are the consistency conditions. These conditions are met in the sense of the collocation (least-squares) method. Numerical examples are considered, including a problem with internal layers, for which the method we suggest yields a good uniform accuracy.Brought to you by | University of Arizona Authenticated Download Date | 5/30/15 9:39 PM

Grid-projection method with small angles in the cells

“…Currently the grid-projection methods on irregular grids have been developed [1, 4,5,7,[12][13][14]17,20]. The problem of constructing an adaptive grid is simplified in them, viz.…”

“…we can use grids with square cells only (cubic cells in the three-dimensional space). A grid-projection method for solving elliptic problems on irregular grids with square cells is suggested in [17]. It allowed us to obtain sufficiently exact numerical solutions of the problems with internal layers on small-size computers, because the adaptive grids contained a considerably fewer number of cells than uniform grids and the method was quite suitable for these grids.…”

“…This is due to the structure of an irregular grid. It cannot become dense abruptly in a neighbourhood of an internal layer or a boundary layer and therefore it becomes denser some distance away from these layers, though the solution changes weakly (see [17] Fig. 9).…”

“…When constructing the grid-projection method on an irregular grid [17] we had to discard the Lagrange representation and the consistency conditions of a collocation type. Thus, we obtained the method whose matrix on a very dense grid has a small condition number.…”

“…When employing the regular grids we could substantially reduce the number of cells as compared to the number of cells in the irregular grids in [17] if we could use highly elongated cells in them. The question is whether it is possible to use highly elongated cells in the grid-projection methods.…”