Hp-Version of the Least-Squares Collocation Method for Elliptic Problems with Second-Order Highest Derivatives on Refined Triangular Grids

A new hp-version of the least-squares collocation method (hp-LSCM) is proposed, implemented, and verified for the numerical solution of elliptic problems with second-order highest derivatives on refined triangular grids generated in Gmsh. The method employs approximation spaces of polynomial degrees p = 2, 3, 5, 6, 8. The unknown coefficients are obtained by solving overdetermined sparse systems of linear algebraic equations using an orthogonal factorization method provided by the SuiteSparse library, combined with CUDA-based parallelization. The developed hp-LSCM is tested on the Dirichlet problem for the Poisson equation in a square domain and on the bending of a clamped annular plate within the Reissner-Mindlin theory. Solutions with large gradients and limited smoothness are considered. A comparison is made with the p=4 version of the collocation method (based on the LSCM), which leads to a square system matrix. Additionally, the conditioning and accuracy of the solutions are analyzed with respect to the number of equations in the approximate problem and the grid refinement parameter.

Keywords

least-squares collocation method; refined triangular grids; Poisson's equation; first-order shear deformation theory; plate bending.

References

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