Multigrid for Locally Refined Meshes

Model case analysis of an algebraic multilevel method

“…A black-box multigrid version, which is related to the method of [25], is introduced and analyzed in [34]. It is further generalized to discretizations based on local mesh refinement in [35], where it is also shown that the condition number of the corresponding two-level method is bounded mesh-and jump-independently.…”

“…In Figure 2, we propose a mesh that is both conformal and regular, using O(| log 2 h|) layers of elements along the discontinuity lines to cause a gradual decrease of resolution across these lines. Once mapped to the isotropic coordinates (that is, (x,ỹ) in the lower left subsquare and (x, y) in the upper right subsquare, wherex = x/8 andỹ = y/8), these elements satisfy the conditions in Section 5 in [35], resulting in a linear finite element scheme whose coefficient matrix is a diagonally dominant M-matrix. Clearly, this scheme involves anisotropy along the piecewise linear curve connecting the points (0.5, 0) to (0.5, 0.5) and (0.5, 0.5) to (1, 0.5), which cannot be resolved by standard coarsening.…”

“…The present analysis is limited to problems that are symmetric positive definite (SPD) with respect to some diagonal inner product. The method is designed so that the two-level analysis of [34,35] is valid, which implies meshand jump-independent condition numbers for model diffusion problems with discontinuous coefficients. This analysis is also extended here to the multilevel case.…”

“…A black-box multigrid version, which is related to the method of [25], is introduced and analyzed in [34]. It is further generalized to discretizations based on local mesh refinement in [35], where it is also shown that the condition number of the corresponding two-level method is bounded mesh-and jump-independently.The above-mentioned methods assume that the linear system of equations arises from a discretization of an elliptic boundary value problem. They inherently use the topology of the domain on which the PDE is defined for constructing the coarse grids.…”

A Parallel Geometric Multigrid Method for Finite Elements on Octree Meshes

“…Multigrid algorithms are known to be efficient for solving elliptic PDEs. They have been researched extensively in the past [7,14,15,22,32,33,50,61,62,63,64] and remain an active research area [1,2,10,11,24,28,32,34]. A distinguishing feature of multigrid algorithms is that their convergence rate does not deteriorate with increasing problem size [18,33,57].…”