The Multi-Grid Method for the Diffusion Equation with Strongly Discontinuous Coefficients

Abstract

The subject of this paper is the application of the multi-grid method to the solution of -V. (D(x, y)VU(x, y))+(x, y)U(x, y)=f (x, y) in a bounded region f of R where D is positive and D, tr, and f are allowed to be discontinuous across internal boundaries F of fL The emphasis is on discontinuities of orders of magnitude in D, when special techniques must be applied to restore the multi-grid method to good efficiency. These techniques are based on the continuity of D VU across F. Two basic methods are derived, one in which the approximating finite difference operators on coarser grids are five point operators (assuming the finite difference operator on the finest grid is a five point one) and one in which they are nine point operators.

An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients

“…Hence a contrast of 10 −7 is common in the system of equations to be solved. Other applications where the coefficients have large discontinuities are electrical power networks [21], groundwater flow [1,18], semiconductors [11], and electromagnetics modeling [19].…”

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

“…This assumption is supported by the experiments that we present in Sect. 5, that show that the costs of factorization are substantially greater than a single iterative solve and, consequently, that iterative solvers can reduce the total required computational time.…”

“…In geometric multigrid schemes, the coarse-grid operators and intergrid transfer operators (interpolation and restriction) are determined based on explicit knowledge of the grid geometry and discretized PDE. In contrast, interpolation operators for AMG are defined in matrix-dependent ways [5,40], while the restriction and coarse-grid operators are given by variational conditions (when K is symmetric and positive definite) [35]. Thus, the challenge in achieving efficient multigrid performance is focused on the definition of appropriate matrix-dependent interpolation operators.…”

Fast and robust solvers for pressure-correction in bubbly flow problems

“…Instead, we aim to improve the multigrid performance by making a different choice for the interpolation operator, Z, to better complement the performance of lexicographical Gauss-Seidel relaxation. It has long been recognized that for problems with discontinuous coefficients, such as (5), the errors left after relaxation are not smooth, as in the case of constant-coefficient problems [2]. Thus, while coarse-grid correction with a fixed interpolation operator, such as those analyzed in [25] and discussed above, may be used to effectively complement relaxation for constant-coefficient problems, they are less appropriate when the problem contains large jumps in its coefficients.…”

“…The black box multigrid technique, first introduced in [2], uses geometrically structured coarse grids in combination with an interpolation operator designed to account for the effects of jumps in the diffusion coefficients to achieve fast multigrid convergence in many situations [4,8,10]. Algebraic multigrid, or AMG, is also known to be effective for elliptic problems with jumps in their coefficients [33,36], achieving this efficiency by tailoring both the coarse-grid structure and interpolation operator to account for the jumps in the coefficients.…”