Determining the law of ice deformation

h-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications

This paper reports the least squares collocation method proposed and implemented for solving boundary value problems for biharmonic-type equations. The solutions of these equations are used to analyze and simulate the bending of isotropic and orthotropic thin plates. To increase the stability of computations by the least squares collocation method it is proposed to integrate collocation equations over the subcells of each cell of a computational grid.

The least squares collocation method with the integral form of collocation equations for bending analysis of isotropic and orthotropic thin plates

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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On the Effective Implementation and Capabilities of the Least-Squares Collocation Method for Solving Second-Order Elliptic Equations

This paper describes new versions of the least-squares collocation method for solving differential and integral equations. A p-version of the method has been proposed and implemented to solve nonlinear systems of partial differential equations. The stationary Navier-Stokes equations are used as an example. A hp-version of the method has been implemented for the numerical solution of the Fredholm integral equations of the second kind in the one-and two-dimensional cases. This paper shows that approximate solutions obtained by various versions of the least-squares collocation method converge with a high order and agree with analytical solutions of test problems with a high degree of accuracy.

https://iopscience.iop.org/article/10.1088/1742-6596/1715/1/012031/pdf

New versions of the least-squares collocation method for solving differential and integral equations

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V. P. Shapeev

Papers

Solving Elliptic Equations in Polygonal Domains by the Least Squares Collocation Method

The least squares collocation method with the integral form of collocation equations for bending analysis of isotropic and orthotropic thin plates

The least squares collocation method for biharmonic-type equations in the plate theory