G.I. Shiskin, P. Hemker

Robust Methods for Problems with Layer Phenomena and Additional Singularities

The minisymposium will be concerned with singularly perturbed multiscale problems having additional singularities. A complicated geometry or unboundedness of the domain and/or the lack of sufficient smoothness (or compatibility) of the problem data may result in singular solutions that have their own specific scales, besides boundary/interior layers. We intend to examine techniques for constructing numerical methods that converge parameter-uniformly (in the maximum norm)

The following research aspects will be also considered: (i) As a rule, such parameter-uniformly convergent numerical methods have too low order of uniform convergence, which restricts their applicability in practice. With this respect, methods how to increase the accuracy of parameter-uniformly convergent numerical methods will be considered. (ii) When standard numerical methods, for example, domain decomposition methods are used to find solutions of parameter-uniformly convergent discrete approximations, the decomposition errors of the discrete solutions and the number of iterations required to solve the discrete problem depend on the perturbation parameter and grow when it tends to zero. We will consider decomposition methods preserving the property of parameter-uniform convergence. Domain decomposition and local defect correction techniques allows us to reduce the construction of robust numerical methods for multiscale problems to locally robust methods for monoscale problems on the specific subdomains. Other aspects and applications will be also under consideration. Problems for partial differential equations with different types of boundary and interior layers will be considered. To construct special numerical methods, fitted meshes, which are a priori and a posteriori condensing in the layer regions, are used.