A geometric‐based algebraic multigrid method for higher‐order finite element equations in two‐dimensional linear elasticity

Abstract

In this paper, we will discuss the geometric-based algebraic multigrid (AMG) method for two-dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and higher-order finite element space. And then a geometric-based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high-order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests.

Semianalytical analysis of shear walls with the use of discrete-continual finite element method. Part 2: Numerical examples, future development

“…Algebraic multigrid (AMG) methods [24][25][26][27] are gaining prominence due to their generality and the ability to deal with unstructured meshes. Geometric multigrid methods are less general but have low overhead, are quite fast, and are easy to parallelize (at least for structured grids) [28][29][30].…”

A Symmetric Finite Volume Element Scheme on Tetrahedron Grids

“…We consider the coefficient K(x) in two cases The results from Table 1 to Table 4 are for the basic convergent estimations in L 2 norm, L ∞ norm and H 1 norm. The PCG methods [3,25,27,28,29] are used to solve the corresponding discrete systems, and improve the efficiency of computation. In tables,…”

A robust preconditioner for higher order finite element discretizations in linear elasticity

“…l = 1, 2) FEM equations (6) is well developed. The approach adopted in this paper is an AMG method that can be regarded as an extension of the algebraic multigrid technique for the solution of linear systems arising from the discretization of scalar elliptic PDE to systems of PDEs, for which the detailed description can be found in [12] and [22]. The construction of prolongations is performed by some energy minimization procedures, see e.g.…”

“…While the AMG method addressed in 18, 19 has several advantages, it may not provide an efficient solver for some cases of higher order elements, and moreover, the corresponding theory is not included in their work. Besides, several efficient AMG methods are developed in 20–22 for higher order discretizations in two and three dimensions by using both geometric and algebraic information. Theoretically, it can be viewed as a two‐level method in which the coarse space is a linear finite element space and its convergence is analyzed by using a general identity presented in 23 for the second‐order elliptic boundary value problem discretized by finite elements on shape‐regular grids.…”