Multigrid finite element methods on semi‐structured triangular grids for planar elasticity

Abstract

SUMMARYMultigrid methods for a stencil-based implementation of a finite element method for planar elasticity, using semi-structured triangular grids, are presented. Local Fourier analysis (LFA) is applied to identify the correct multigrid components. To this end, LFA for multigrid methods on regular triangular grids is extended to the case of the problem of planar elasticity, although its application to other systems is straightforward. For the discrete elasticity operator obtained with linear finite elements, different collective smoothers, such as three-color smoother and some zebra-type smoothers, are analyzed, and LFA results for these smoothers are shown. The multigrid method is constructed in a block-wise form. In particular, different smoothers and different numbers of pre-and post-smoothing steps are considered in each triangle of the coarsest triangulation of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm.

On a local Fourier analysis for overlapping block smoothers on triangular grids

“…LFA was generalized to triangular grids in [7], for discretizations based on linear finite element methods. Afterwards, this generalization has been extended to systems of partial differential equations [8,9] and to high-order finite element discretizations [25].…”

Poroelasticity problem: numerical difficulties and efficient multigrid solution

“…To confirm these unstable behaviour, we solve system (13) in the computational domain (0, 1) by using linear finite elements considering K E τ = 10 −6 . In this case, it is necessary a mesh of at least 500 nodes to fulfill the restriction.…”

“…LFA was generalized to triangular grids in [10], for discretizations based on linear finite element methods. Afterwards, we extended this generalization to systems of partial differential equations [12,13].…”

Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis

“…We will use the block-wise processing to choose different smoothers depending on the shape of the tetrahedra. This kind of strategy has already been used successfully for two-dimensional linear-elasticity problems in [14]. Here, we will apply it for three-dimensional tetrahedral meshes.…”