Adaptive Discontinuous Galerkin Methods with Applications

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Many problems of engineering interest are posed in complex, possibly non-smooth and three-dimensional domains. They often involve multiple physical models and scales, as well as highly varying or non-linear material properties. Advanced numerical techniques are required for meaningful predictions. To achieve improved and high-order accurate simulation outputs, we shall focus on the development and implementation of discontinuous Galerkin (DG) finite element methods, combined with adaptive refinement strategies. In recent years, these methods established themselves as robust and flexible simulation techniques for wide classes of problems in computational fluid dynamics, structural mechanics and electro-magnetics. They are naturally suited for adaptive computations, and for providing a unified computational platform for simulating coupled multi-physics models in three space dimensions. The objectives of this proposal are twofold. First, we plan to design and analyse adaptive discontinuous Galerkin (DG) finite element methods for the numerical approximation of partial differential equations of various types in three space dimensions. We are particularly interested in realizing fully automatic refinement strategies, which allow for anisotropic refinements in both the elemental mesh sizes and approximation degrees, and which yield high-order or exponential convergence rates in the numbers of degrees of freedom, also when singularities or layers are present in the phenomena of interest. In addition, we wish to develop fast and efficient spectral/hp element domain decomposition solvers. The second objective is the further advancement and efficient implementation of these methods as applied to coupled incompressible fluid flow problems. We intend to continue our work on fast solvers for electrically conducting fluids interacting with magnetic fields, and to investigate similar techniques for the computer simulation of non-isothermal incompressible flow problems.