Algebraic Multigrid (AMG) for indefinite linear systems from meshfree dscretizations

Koung-Hee Leem

Department of Mathematics, 14 MacLean Hall, University of Iowa, Iowa City, IA 52242-1419

S. Oliveira
D. E. Stewart


Meshfree discretizations construct approximate solutions to partial differential equation based on particles, not on meshes, so that it is well suited to solve the problems on irregular domains. Since the nodal basis property is not satisfied in meshfree discretizations, it is difficult to handle essential boundary conditions. There has been attempts to enforce this property by modifying basis functions. Instead, we employ the Lagrange multiplier approach to resolve this problem, but this will result in indefinite linear systems. As a solver for this indefinite linear system, we propose a new Algebraic Multigrid (AMG) scheme based on aggregating elements and neighborhoods. AMG based only on aggregation of elements was tried because of its simplicity, but it often shows slow convergence. There have been many attempts to overcome this problem. The new interpolation approach utilizes the information in neighborhood matrices as well as aggregation of elements. Unlike classical AMG, this new approach can be applied to any symmetric, positive definite linear systems. Moreover, with additional information for meshfree discretizations, we successfully modified this new AMG approach to solve symmetric indefinite linear systems arising from meshfree discretizations with essential boundary conditions.