hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

Abstract

a b s t r a c tThe 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 steady state and time-dependent problems, and low and high cell Reynolds numbers, including highly stretched meshes.

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

“…Furthermore, the Laplacian operator is denoted as M, the initial flow field by u 0 and the boundary data by u b . The details of the space-time discontinuous Galerkin discretization for the advection-diffusion equation can be found in Part I [16].…”

“…In this section we summarize the hp-Multigrid as Smoother algorithm, which we presented in Part I [16] as a new multigrid algorithm for the solution of algebraic systems resulting from higher order accurate finite element discretizations of partial differential equations. The hp-MGS algorithm consists of three steps.…”

“…In addition, we require that each of the semi-coarsening multigrid algorithms HS i nh;p ; i ¼ 1; 2, given by Algorithm 3, have a spectral radius less than one. The operator norms and spectral radii of the multigrid error transformation operator and the Runge-Kutta smoothers are computed for the two-dimensional advectiondiffusion equation using the discrete Fourier multilevel analysis discussed in Part I, [16].…”

“…In this article we only consider steady state problems, since it is considerably more difficult to obtain good multigrid performance for steady state than for time-accurate problems with a space-time DG discretization. For more details, see Part I, [16]. The steady state solution of the advection-diffusion equation depends on the cell Reynolds number Re h , the mesh aspect ratio and the flow angle, see Section 2.…”

“…In [16], subsequently called Part I, we introduced the hp-Multigrid as Smoother algorithm as a new multigrid method for the solution of algebraic systems resulting from higher order accurate finite element discretizations of partial differential equations. Using discrete Fourier multilevel analysis the multigrid performance of the full hp-MGS algorithm was analyzed for two-dimensional problems.…”