Geodesic
The hp-version of the least-squares collocation method with~integral collocation for solving a biharmonic equation
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 3, pp. 556-572

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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.
Keywords: least-squares collocation method, integral collocation equation, biharmonic equation, plate bending, acceleration of iterative processes, preconditioning, Krylov subspaces, multigrid algorithms, parallelization. 
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     author = {V. P. Shapeev and L. S. Bryndin and V. A. Belyaev},
     title = {The hp-version of the least-squares collocation method with~integral collocation for solving a biharmonic equation},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {556--572},
     publisher = {mathdoc},
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     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2022_26_3_a7/}
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V. P. Shapeev; L. S. Bryndin; V. A. Belyaev. The hp-version of the least-squares collocation method with~integral collocation for solving a biharmonic equation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 3, pp. 556-572. http://geodesic.mathdoc.fr/item/VSGTU_2022_26_3_a7/