Numerical Analysis Seminars: High-Order/hp-Adaptive Multilevel Discontinuous Galerkin Methods
6 December 2013 14:00 in CM105
We present a discontinuous Galerkin (DG) multilevel method with hp-adaptivity. The main advantage of this multilevel method is that the number of dimensions of the finite element space is independent on the presence of complicated or tiny features in the domain. In other words, even on a very complicated domain, an approximation of the solution can be computed with only a fistful of degrees of freedom. This is possible because two meshes are used: a fine mesh is used to describe the geometry of the domain with all its features, but the problem is actually solved on a coarse mesh that is, in general, too coarse to describe all the geometrical features of the domain. Unlikely other multilevel methods, this method does not perturb the problem, in the sense that the problem solved on the coarse mesh is always a discretization of the continuous problem, no matter how coarse the mesh is. The method itself is a hp-adaptive DG extension of composite finite elements (CFEs), introduced by S. Sauter a few years ago.
Standard CFE methods are based on standard continuous Galerkin elements, which means that there are restrictions on the kind of boundary conditions that can be used. These limitations disappear by extending the method to DG elements.
The hp-adaptivity algorithm that we present for this multilevel method is completely automatic and capable of exploiting both local polynomial-degree-variation (p-refinement) and local mesh subdivision (h-refinement), thereby offering greater flexibility and efficiency than numerical techniques which only incorporate h-refinement or p-refinement alone.
We employed two types of error estimators: an explicit one and an implicit one based on a duality argument. In the latter case the element residuals of the computed numerical solution are multiplied by local weights involving the solution of a certain dual or adjoint problem. On the basis of the resulting a posteriori error bounds, we implemented an adaptive finite element algorithm to ensure reliable and efficient control of the error. Our adaptive finite element algorithm decides automatically between either h-refine or p-refine. The performance of the resulting hp-refinement algorithm is demonstrated through a series of numerical experiments.
Moreover, the projection operator between the two levels, which is used to move the problem from the fine level mesh to the coarse level mesh, can be applied also to construct multi-levels Schwarz preconditioners for complicated domains where the coarse level mesh is too coarse to describe all the geometrical features.
This research has been funded by the EPSRC.
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