Algebraic multigrid for systems of elliptic boundary‐value problems

Abstract

SummaryThis article develops an algebraic multigrid (AMG) method for solving systems of elliptic boundary‐value problems. It is well known that multigrid for systems of elliptic equations faces many challenges that do not arise for most scalar equations. These challenges include strong intervariable couplings, multidimensional and possibly large near‐nullspaces, analytically unknown near‐nullspaces, delicate selection of coarse degrees of freedom (CDOFs), and complex construction of intergrid operators. In this article, we consider only the selection of CDOFs and the construction of the interpolation operator. The selection is an extension of the Ruge–Stuben algorithm using a new strength of connection measure taken between nodal degrees of freedom, that is, between all degrees of freedom located at a gridpoint to all degrees of freedom at another gridpoint. This measure is based on a local correlation matrix generated for a set of smoothed test vectors derived from a relaxation‐based procedure. With this measure, selection of the CDOFs is then determined by the number of strongly correlated connections at each node, with the selection processed by a Ruge–Stuben coloring scheme. Having selected the CDOFs, the interpolation operator is constructed using a bootstrap AMG (BAMG) procedure. We apply the BAMG procedure either over the smoothed test vectors to obtain an intervariable interpolation scheme or over the like‐variable components of the smoothed test vectors to obtain an intravariable interpolation scheme. Moreover, comparing the correlation measured between the intravariable couplings with the correlation between all couplings, a mixed intravariable and intervariable interpolation scheme is developed. We further examine an indirect BAMG method that explicitly uses the coefficients of the system operator in constructing the interpolation weights. Finally, based on a weak approximation criterion, we consider a simple scheme to adapt the order of the interpolation (i.e., adapt the caliber or maximum number of coarse‐grid points that a fine‐grid point can interpolate from) over the computational domain.