Distributive smoothers in multigrid for problems with dominating grad–div operators

Abstract

SUMMARYIn this paper we present efficient multigrid methods for systems of partial differential equations that are governed by a dominating grad-div operator. In particular, we show that distributive smoothing methods give multigrid convergence factors that are independent of problem parameters and of the mesh sizes in space and time. The applications range from model problems to secondary consolidation Biot's model. We focus on the smoothing issue, and mainly solve academic problems on Cartesian staggered grids.

Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs

“…Very similar results are obtained with finite-difference multigrid experiments for the grad-div operator in [29]. These results can be labeled as textbook multigrid efficiency, and an increasing number of smoothing steps further decreases the two-grid factors.…”

“…While the use of these smoothers leads to efficient multigrid approaches for systems of PDEs, collective relaxation is not the only possible approach. The main alternative is the use of distributive smoothers [26,28,29], which take their name from a distribution operation; the discrete (or continuum) equations are transformed by right matrix multiplication into a block triangular matrix that is amenable to pointwise relaxation. Simple pointwise relaxation is performed on this block triangular system and, then, the resulting update is distributed (based on the transformation matrix) back to the original matrix problem.…”

“…While distributed smoothers are often found to be more efficient than overlapping smoothers [29], their applicability is limited by the need to find an effective distribution matrix; this is often difficult to do for problems with unstructured grids or variable coefficients. Thus, distributed relaxation may be difficult to implement in a general and purely algebraic fashion, as would be necessary for use within an algebraic multigrid iteration.…”

“…These operators appear frequently in the formulation of mathematical models in physics and engineering, particularly for problems related to electro-magnetics or fluid and solid mechanics (see, for example, [15,29,33] for more details).…”

A second-order virtual node algorithm for nearly incompressible linear elasticity in irregular domains

“…Many authors have also looked at applying multigrid methods to problems in solid mechanics (e.g. [1,[51][52][53][54][55][56][57][58][59][60][61][62]), including handling issues arising from nearly incompressible materials. Mixed FEM formulations are one example that maintain good multigrid convergence properties for nearly incompressible materials demonstrated on the pure Dirichlet boundary case [54,58,61].…”