Research Problems

1. Grid generation

Grid generation consistent with finite difference/volume discretizations. The main problem under study is how the grid and the method of constructing of finite difference/volume schemes interact and affect the accuracy in the approximate solution. It is important that transformed equation contains derivatives of functions, which determine transformation, and to descretize the transformed equation we also need to descretize these derivatives. It means that we assume that these derivatives exist and also some more higher derivatives of mapping exist and bounded to ensure second order formal accuracy. If grid is actually does not satisfy such properties then it can lead to decreasing of accuracy.

Some algorithm requires that the differential equation be written in terms of a time derivative, divergence, gradient, and curl. Then the individual first-order differential operators are discretized and then these discretizations are used to discretize the differential equation. When deriving finite-difference scheme by support-operators method we do not assume that grid is obtained by smooth mapping. Then we can anticipate that this method will work better for unsmooth grid.

2. Mimetic discretizations with rough coefficients

3. Mimetic discretizations on rough grids

Here the grid is given and it is not in general a smooth grid so the discretization has to be able to work on rough grids as well as in this rough grids.

4. High Order Mimetic Finite Difference/Volume Methods

Difference approximations that retain the symmetry properties of the continuum operators are called mimetic Partial differential equations solved with mimetic difference approximations often automatically satisfy discrete versions of conservation laws and analogies to Stoke's theorem that are true in the continuum and therefore are more likely to produce physically faithful results. These symmetries are easily preserved by local discrete high-order approximations on uniform grids, but are difficult to retain in high-order approximations on nonuniform grids. The main goal of this research is to construct local high-order mimetic difference approximations of differential operators on nonuniform grids. Local approximations only use function values at nearby points in the computational grid and are especially efficient on computers with distributed memory. High-order approximations can often be used to solve
partial differential equation (PDEs) to a prescribed accuracy with only a fraction of the grid points that would be required by a first or second order method.