PREFACE

Many real-world problems are described by partial differential equations whose numerical solution represents an important part of numerical mathematics. There are several techniques for their solution: the finite difference method, the finite element method, spectral methods and the finite volume method. All these methods have advantages as well as disadvantages. The first three techniques are suitable particularly for problems in which the exact solution is sufficiently regular. The presence of interior and boundary layers appearing in solutions of singularly perturbed problems (e.g., convection-diffusion problems with dominating convection) or discontinuities in solutions of nonlinear hyperbolic equations lead to some difficulties. On the other hand, finite volume techniques based on discontinuous, piecewise constant approximations are very useful in solving convection-diffusion problems, but their disadvantage is their low order of accuracy. The most recent technique for the numerical solution of partial differential equations is the discontinuous Galerkin method (DGM), which uses ideas of both the finite element and finite volume methods. The DGM is based on piecewise polynomial but discontinuous approximations, which provides robust numerical processes and high-order accurate solutions. During the past two decades the DGM has become very popular and a number of works has been concerned with its analysis and applications. It appeared that the DGM is suitable for the numerical solution of a number of problems for which other techniques fail or have difficulties. We can mention singularly perturbed problems with boundary and internal layers, which exist in solutions of convection-diffusion equations with dominating convection. Another possibility represents problems with solutions containing discontinuities and steep gradients, as in the case of nonlinear hyperbolic problems and compressible flow. This means that the DGM is suitable for the numerical solution of problems appearing particularly in fluid dynamics, hydrology, heat and mass transfer and environmental protection on the one hand, but also financial mathematics and image processing on the other hand. Moreover, the DGM offers considerable flexibility in the choice of the mesh design; indeed, the DGM easily handles non-matching and non-uniform grids, even anisotropic, with different