HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis

The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time discontinuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes

Local Fourier analysis of multigrid for hybridized and embedded discontinuous Galerkin methods

In this paper we present a geometric multigrid method with Jacobi and Vanka relaxation for hybridized and embedded discontinuous Galerkin discretizations of the Laplacian. We present a local Fourier analysis (LFA) of the two-grid error-propagation operator and show that the multigrid method applied to an embedded discontinuous Galerkin (EDG) discretization is almost as efficient as when applied to a continuous Galerkin discretization. We furthermore show that multigrid applied to an EDG discretization outperforms multigrid applied to a hybridized discontinuous Galerkin (HDG) discretization. Numerical examples verify our LFA predictions

Two-level multigrid analysis for the convection-diffusion equation discretized by a discontinuous Galerkin method.

Keywords continuous Galerkin method ? multigrid iteration ? two-level Fourier analysis ? point-wise block-relaxation Abstract In this paper, we study a multigrid (MG) method for the solution of a linear one-dimensional convection-diffusion equation that is discretized by a discontinuous Galerkin method. In particular we study the convection-dominated case when the perturbation parameter, i.e. the inverse cell-Reynolds-number, is smaller than the finest mesh size. We show that, if the diffusion term is discretized by the non-symmetric interior penalty method (NIPG) with feasible penalty term, multigrid is sufficient to solve the convection-diffusion or the convection-dominated equation. Then, independent of the mesh-size, simple MG cycles with symmetric Gauss-Seidel smoothing give an error reduction factor of 0.2-0.3 per iteration sweep. Without penalty term, for the Baumann-Oden (BO) method we find that only a robust (i.e. cell-Reynolds-number uniform) two-level error-reduction factor (0.4) is found if the point-wise block-Jacobi smoother is used. Copyright ? 2005 John Wiley & Sons, Ltd

Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods

In this thesis we study efficient solvers for space-time discontinuous Galerkin spectral element methods (DG-SEM). These discretizations result in fully implicit schemes of variable order in both spatial and temporal directions. The popularity of space-time DG methods has increased in recent years and entropy stable space-time DG-SEM have been constructed for conservation laws, making them interesting for these applications. The size of the nonlinear system resulting from differential equations discretized with space-time DG-SEM is dependent on the order of the method, and the corresponding Jacobian is of block form with dense blocks. Thus, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption. The lack of good solvers for three-dimensional DG applications has been identified as one of the major obstacles before high order methods can be adapted for industrial applications.It has been proven that DG-SEM in time and Lobatto IIIC Runge-Kutta methods are equivalent, in that both methods lead to the same discrete solution. This allows to implement space-time DG-SEM in two ways: Either as a full space-time system or by decoupling the temporal elements and using implicit time-stepping with Lobatto IIIC methods. We compare theoretical properties and discuss practical aspects of the respective implementations.When considering the full space-time system, multigrid can be used as solver. We analyze this solver with the local Fourier analysis, which gives more insight into the efficiency of the space-time multigrid method. The other option is to decouple the temporal elements and use implicit Runge-Kutta time-stepping methods. We suggest to use Jacobian-free Newton-Krylov (JFNK) solvers since they are advantageous memory-wise. An efficient preconditioner for the Krylov sub-solver is needed to improve the convergence speed. However, we want to avoid constructing or storing the Jacobian, otherwise the favorable memory consumption of the JFNK approach would be obsolete. We present a preconditioner based on an auxiliary first order finite volume replacement operator. Based on the replacement operator we construct an agglomeration multigrid preconditioner with efficient smoothers using pseudo time integrators. Then only the Jacobian of the replacement operator needs to be constructed and the DG method is still Jacobian-free. Numerical experiments for hyperbolic test problems as the advection, advection-diffusion and Euler equations in several dimensions demonstrate the potential of the new approach

Multigrid methods for Maxwell\u27s equations

In this work we study finite element methods for two-dimensional Maxwell\u27s equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell\u27s equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments