# Past Computational Mathematics and Applications Seminar

Domain decomposition methods were introduced as techniques for solving partial differential equations based on a decomposition of the spatial domain of the problem into several subdomains. The initial equation restricted to the subdomains defines a sequence of new local problems. The main goal is to solve the initial equation via the solution of the local problems. This procedure induces a dimension reduction which is the major responsible of the success of such a method. Indeed, one of the principal motivations is the formulation of solvers which can be easily parallelized.

In this presentation we shall develop a domain decomposition algorithm to the minimization of functionals with total variation constraints. In this case the interesting solutions may be discontinuous, e.g., along curves in 2D. These discontinuities may cross the interfaces of the domain decomposition patches. Hence, the crucial difficulty is the correct treatment of interfaces, with the preservation of crossing discontinuities and the correct matching where the solution is continuous instead. I will present our domain decomposition strategy, including convergence results for the algorithm and numerical examples for its application in image inpainting and magnetic resonance imaging.

ALE formulations are useful when approximating solutions of problems in deformable domains, such as fluid-structure interactions. For realistic simulations involving fluids in 3d, it is important that the ALE method is at least of second order of accuracy. Second order ALE methods in time, without any constraint on the time step, do not exist in the literature and the role of the so-called geometric conservation law (GCL) for stability and accuracy is not clear. We propose discontinuous Galerkin (dG) methods of any order in time for a time dependent advection-diffusion model problem in moving domains. We prove that our proposed schemes are unconditionally stable and that the conservative and non conservative formulations are equivalent. The same results remain true when appropriate quadrature is used for the approximation of the involved integrals in time. The analysis hinges on the validity of a discrete Reynolds' identity and generalises the GCL to higher order methods. We also prove that the computationally less intensive Runge-Kutta-Radau (RKR) methods of any order are stable, subject to a mild ALE constraint. A priori and a posteriori error analysis is provided. The final estimates are of optimal order of accuracy. Numerical experiments confirm and complement our theoretical results.

This is joint work with Andrea Bonito and Ricardo H. Nochetto.

The Discontinuous Galerkin (DG) method has been used to solve a wide range of partial differential equations. Especially for advection dominated problems it has proven very reliable and accurate. But even for elliptic problems it has advantages over continuous finite element methods, especially when parallelization and local adaptivity are considered.

In this talk we will first present a variation of the compact DG method for elliptic problems with varying coefficients. For this method we can prove stability on general grids providing a computable bound for all free parameters. We developed this method to solve the compressible Navier-Stokes equations and demonstrated its efficiency in the case of meteorological problems using our implementation within the DUNE software framework, comparing it to the operational code COSMO used by the German weather service.

After introducing the notation and analysis for DG methods in Euclidean spaces, we will present a-priori error estimates for the DG method on surfaces. The surface finite-element method with continuous ansatz functions was analysed a few years ago by Dzuik/Elliot; we extend their results to the interior penalty DG method where the non-smooth approximation of the surface introduces some additional challenges.