Dominik Schotzau

Professor
 

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
Mixed discontinuous Galerkin finite element methods for incompressible magnetohydrodynamics (2011)

We develop and analyze mixed discontinuous Galerkin finite element methods for the numerical approximationof incompressible magnetohydrodynamics problems.Incompressible magnetohydrodynamics is the area of physics that isconcerned with the behaviour of electrically conducting,resistive, incompressible and viscous fluids in the presence ofelectromagnetic fields. It is modelledby a system of nonlinear partial differential equations, which couples theNavier-Stokes equations with the Maxwell equations.In the first part of this thesis, we introduce an interior penalty discontinuousGalerkin method for the numerical approximation of a linearized incompressible magnetohydrodynamicsproblem. The fluidunknowns are discretized with the discontinuous℘k-℘k-1 element pair, whereas the magneticvariables are approximated by discontinuous ℘k-℘k+1 elements. Under minimal regularityassumptions, we carry out a complete a priori error analysis andprove that the energy norm error is optimally convergent in themesh size in general polyhedral domains, thus guaranteeing thenumerical resolution of the strongest magnetic singularities innon-convex domains.In the second part of this thesis, we propose and analyze a new mixed discontinuous Galerkin finite element method for the approximation of a fully nonlinearincompressible magnetohydrodynamics model. The velocity field isnow discretized by divergence-conforming Brezzi-Douglas-Marinielements, and the magnetic field by curl-conformingNédélec elements. In addition to correctly capturing magneticsingularities, the method yields exactly divergence-freevelocity approximations, and is thus energy-stable. We show that the energy norm error isconvergent in the mesh size in possibly non-convex polyhedra, andderive slightly suboptimal a priori error estimates under minimal regularity and small dataassumptions.Finally, in the third part of this thesis, we present two extensions of our discretization techniquesto time-dependent incompressiblemagnetohydrodynamics problems and to Stokes problems with nonstandardboundary conditions.All our discretizations and theoretical results are computationallyvalidated through comprehensive sets of numerical experiments.

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Numerical solution of the time-harmonic Maxwell equations and incompressible magnetohydrodynamics problems (2011)

The goal of this thesis is to develop efficient numerical solvers for the time-harmonicMaxwell equations and for incompressible magnetohydrodynamics problems.The thesis consists of three components. In the first part, we present a fullyscalable parallel iterative solver for the time-harmonic Maxwell equations in mixedform with small wave numbers. We use the lowest order Nedelec elements of thefirst kind for the approximation of the vector field and standard nodal elements forthe Lagrange multiplier associated with the divergence constraint. The correspondinglinear system has a saddle point form, with inner iterations solved by preconditioned conjugate gradients. We demonstrate the performance ofour parallel solver on problems with constant and variable coefficients with up toapproximately 40 million degrees of freedom. Our numerical results indicate verygood scalability with the mesh size, on uniform, unstructured and locally refinedmeshes.In the second part, we introduce and analyze a mixed finite element method forthe numerical discretization of a stationary incompressible magnetohydrodynamicsproblem, in two and three dimensions. The velocity field is discretized usingdivergence-conforming Brezzi-Douglas-Marini (BDM) elements and the magneticfield is approximated by curl-conforming Nedelec elements. Key features of themethod are that it produces exactly divergence-free velocity approximations, andthat it correctly captures the strongest magnetic singularities in non-convex polyhedraldomains. We prove that the energy norm of the error is convergent in themesh size in general Lipschitz polyhedra under minimal regularity assumptions,and derive nearly optimal a-priori error estimates for the two-dimensional case. Wepresent a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for two-dimensional as well as three-dimensionalproblems.Finally, in the third part we investigate preconditioned Krylov iterations forthe discretized stationary incompressible magnetohydrodynamics problems. Wepropose a preconditioner based on efficient preconditioners for the Maxwell andNavier-Stokes sub-systems. We show that many of the eigenvalues of the preconditionedsystem are tightly clustered, and hence, rapid convergence is accomplished.Our numerical results show that this approach performs quite well.

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Robust a posteriori error estimation for discontinuous Galerkin methods for convection diffusion problems (2010)

The present thesis is concerned with the development and practical implementation of robust a-posteriori error estimators for discontinuous Galerkin (DG) methods for convection-diffusion problems.It is well-known that solutions to convection-diffusion problems may have boundary and internal layers of small width where their gradients change rapidly. A powerful approach to numerically resolve these layers is based on using hp-adaptive finite element methods, which control and minimize the discretization errors by locally adapting the mesh sizes (h-refinement) and the approximation orders (p-refinement) to the features of the problems. In this work, we choose DG methods to realize adaptive algorithms. These methods yield stable and robust discretization schemes for convection-dominated problems, and are naturally suited to handle local variations in the mesh sizes and approximation degrees as required for hp-adaptivity.At the heart of adaptive finite element methods are a-posteriori error estimators. They provide information on the errors on each element and indicate where local refinement/derefinement should be applied. An efficient error estimator should always yield an upper and lower bound of the discretization error in a suitable norm. For convection-diffusion problems, it is desirable that the estimator is also robust, meaning that the upper and lower bounds differ by a factor that is independent of the mesh Peclet number of the problem.We develop a new approach to obtain robust a-posteriori error estimates for convection-diffusion problems for h-version and hp-version DG methods. The main technical tools in our analysis are new hp-version approximation results of an averaging operator, which are derived for irregular hexahedral meshes in three dimensions, as well as for irregular anisotropic rectangular meshes in two dimensions.We present a series of numerical examples based on C++ implementations of our methods. The numerical results indicate that the error estimator is effective in locating and resolving interior and boundary layers. For the hp-adaptive algorithms, once the local mesh size is of the same order as the width of boundary or interior layers, both the energy error and the error estimator are observed to converge exponentially fast in the number of degrees of freedom.

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Master's Student Supervision (2010 - 2018)
An exactly divergence-free finite element method for non-isothermal flow problems (2013)

In this thesis the exactly divergence-free finite element method developed by Cockburn, Scheotzau and Kanschat in [13] and [14] is studied. This method is first reviewed in the context of Stokes problem. An interiorpenalty discontinuous Galerkin approach is formulated and analysed in the unified framework established in[13], [14] and [37]. Then we extend the method to non-isothermal flow problems, in particular, to a generalised Boussinesq equation. Following the work by Ricardo, Scheotzau and Qin in [34], the method is formulated andthe numerical analysis is reviewed. Numerical examples are implemented and presented,which verify the theoretical error estimates and the exactly divergence-free property.

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