Solving Negative Order Equations by the Multigrid Method Via Variable Substitution

Variable substitutions are introduced to the single layer potential equations such that the order of pseudo-differential operator is changed from minus one to plus one. Though the condition number remains the same order after such a variable substitution, the frequencies of higher and lower eigenfunctions are switched. The multigrid iteration is shown to be an optimal order solver for the resulting linear systems of boundary element equations. Two types of variable substitutions are suggested. Numerical tests are presented showing efficiency of both methods, and supporting the theory.

[1]  Jinchao Xu,et al.  The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems , 1988 .

[2]  Martin Costabel,et al.  On boundary integral equations of the first kind for the bi-laplacian in a polygonal plane domain , 1983 .

[3]  Martin Kuefer.  Natural Boundary Integral Method And Its Applications , 2016 .

[4]  Shangyou Zhang.  OPTIMAL-ORDER NONNESTED MULTIGRID METHODS FOR SOLVING FINITE ELEMENT EQUATIONS I: ON QUASI-UNIFORM MESHES , 2010 .

[5]  George C. Hsiao,et al.  Optimal order multigrid methods for solving exterior boundary value problems , 1994 .

[6]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[7]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[8]  J. Pasciak,et al.  The analysis of multigrid algorithms for pseudodifferential operators of order minus one , 1994 .

[9]  Randolph E. Bank,et al.  An optimal order process for solving finite element equations , 1981 .

[10]  Shangyou Zhang,et al.  On the convergence of spectral multigrid methods for solving periodic problems , 1991 .

[11]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[12]  Wolfgang L. Wendland,et al.  Some applications of a galerkin‐collocation method for boundary integral equations of the first kind , 1984 .

[13]  Shangyou Zhang,et al.  Optimal-order nonnested multigrid methods for solving finite element equations III: on degenerate meshes , 1995 .

[14]  W. S. Venturini.  Boundary Integral Equations , 1983 .

[15]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[16]  G. C. Hsiao,et al.  Boundary Element Methods: Foundation and Error Analysis , 2004 .