General interior Hermite collocation methods for second-order elliptic partial differential equations
The paper discusses the iterative solution of linear systems resulting from the collocation based on \(C^ 1\)-piecewise polynomials for second- order elliptic partial differential equations on rectilinear regions. First it describes the code GENCOL for general cubic Hermit collocation in 2D domains and the simplified code GINCOL for rectilinear domains and uncoupled boundary conditions. Since the properties of the resulting collocation matrix strongly depends on the ordering of the unknowns and equations, two kinds of ordering schemes are discussed, the finite element and the tensor product ordering. It is shown that tensor product ordering produces linear systems suitable for block iterative solvers of overrelaxation and conjugate gradient type. A number of numerical experiments are reported that indicate the applicability and effectiveness of accelerated overrelaxation and conjugate gradient iterative schemes.
second-order elliptic partial differential equations
scientific article; zbMATH DE number 721001
scientific article