DOI: https://doi.org/10.26089/NumMet.v22r313

On the effective implementation and capabilities of the least-squares collocation method for solving second-order elliptic equations

Authors

  • V.A. Belyaev

Keywords:

least-squares collocation method
high order of accuracy
Poisson equation
diffusion-convection equation
large gradients
discontinuity of solution
preconditioning
Krylov subspaces
multigrid complex
parallelization

Abstract

The capabilities of the numerical least-squares collocation (LSC) method of the piecewise polynomial solution of the Dirichlet problem for the Poisson and diffusion-convection equations are investigated. Examples of problems with singularities such as large gradients and discontinuity of the solution at interfaces between two subdomains are considered. New hp-versions of the LSC method based on the merging of small and/or elongated irregular cells to neighboring independent cells inside the domain are proposed and implemented. They cut off by a curvilinear interface from the original rectangular grid cells. Taking into account the problem singularity the matching conditions between the pieces of the solution in cells adjacent from different sides to the interface are written out. The results obtained by the LSC method are compared with other high-accuracy methods. Advantages of the LSC method are shown. For acceleration of an iterative process modern algorithms and methods are applied: preconditioning, properties of the local coordinate system in the LSC method, Krylov subspaces; prolongation operation on a multigrid complex; parallelization. The influence of these methods on iteration numbers and computation time at approximation by polynomials of various degrees is investigated.

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Published

2021-09-14

Issue

Vol. 22 (2021): Issue 3.

Section

Methods and algorithms of computational mathematics and their applications

Author Biography

V.A. Belyaev

S.A. Khristianovich Institute of Theoretical and Applied Mechanics of SB RAS

• Junior Researcher

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