The hp-version of the least-squares collocation method with integral collocation for solving a biharmonic equation

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Abstract

A new algorithm for the numerical solution of the biharmonic equation is developed. It is based on the first implemented hp-version of the least-squares collocation method (hp-LSCM) with integral collocations for a fourth-order elliptic equation in combination with modern methods of accelerating iterative processes for solving systems of linear algebraic equations (SLAE). The hp-LSCM provides the possibilities to refine the grid (h-version) and increase the degree of polynomials to the arbitrary order (p-approach). The convergence of approximate solutions obtained by the implemented version of the method is analyzed using an example of a numerical simulation of the bending of a hinged isotropic plate. The high accuracy and the increased order of convergence using polynomials up to the tenth order in the hp-LSCM are shown.

The effectiveness of the combined application of algorithms for accelerating iterative processes to solve SLAE that are combined with LSCM is investigated. In this paper, we consider the application of the following algorithms: preconditioning of SLAE matrices; the iteration acceleration algorithm based on Krylov subspaces; the prolongation operation on a multigrid complex; parallelization using OpenMP; a modified algorithm for solving local SLAEs. The latter is implemented with iterations over subdomains (which are cells) and makes it possible to more effectively solve overdetermined SLAEs in the LSCM in the case of solving a linear differential equation. The form of the matrices does not change at each iteration. Only the elements of the vectors of their right parts corresponding to the matching conditions are modified. The calculation time on a personal computer is reduced by more than 350 times with the combined use of all acceleration techniques compared to the case when only preconditioning was used.

About the authors

Vasily P. Shapeev

Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences;
Novosibirsk State University

Email: shapeev.vasily@mail.ru
ORCID iD: 0000-0001-6761-7273
SPIN-code: 7128-5536
Scopus Author ID: 8948291400
ResearcherId: J-7816-2018
http://www.mathnet.ru/person48027

Dr. Phys. & Math. Sci., Professor; Chief Researcher; Lab. of Thermomechanics and Strength of New Materials1; Professor; Dept. of Mathematical Modeling2

4/1, Institutskaya st., Novosibirsk, 630090, Russian Federation; 1, Pirogova st., Novosibirsk, 630090, Russian Federation

Luka S. Bryndin

Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences;
Novosibirsk State University

Email: l.bryndin@g.nsu.ru
ORCID iD: 0000-0002-0211-5800
SPIN-code: 4155-0327
Scopus Author ID: 57209026180
ResearcherId: U-5845-2018
http://www.mathnet.ru/person151108

Junior Research Scientist; Lab. of Thermomechanics and Strength of New Materials1; Postgraduate Student; Dept. of Mathematical Modeling2

4/1, Institutskaya st., Novosibirsk, 630090, Russian Federation; 1, Pirogova st., Novosibirsk, 630090, Russian Federation

Vasily A. Belyaev

Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences

Author for correspondence.
Email: belyaevasily@mail.ru
ORCID iD: 0000-0001-5901-2880
SPIN-code: 3276-0460
Scopus Author ID: 57197726181
ResearcherId: L-2637-2016
http://www.mathnet.ru/person132321

Junior Research Scientist; Lab. of Thermomechanics and Strength of New Materials1

Russian Federation, 4/1, Institutskaya st., Novosibirsk, 630090, Russian Federation

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