New Possibilities and Applications of the Least Squares Collocation Method

Abstract

Abstract. Least squares collocation method (LSC) is a versatile numerical method for solving boundary value problems for PDE. The present article demonstrates the abilities of LSC to solve various problems -in particular, calculations of bending of isotropic irregular shaped plates and multi-layered anisotropic plates. In order to achieve higher accuracy, new versions of the method utilize high-degree polynomial spaces. The numerical experiments demonstrate high accuracy of the solutions.

Collocation and Least Residuals Method and Its Applications

“…Various versions of the CLS and CLR methods have been tested against such benchmarks [9][10][11][12][13][14]. For simplicity, the following discussion is confined to a square cavity…”

“…For different versions of the CLR method for solving the 2D and 3D Navier-Stokes equations, the acceleration algorithm of the iteration processes based on Krylov subspaces [10,11,14,20] was also used with the aim at reducing the CPU time. Its application on a grid of the 256 × 256 size in problems with the Reynolds number Re = 1000 sometimes resulted in an acceleration of the CPU time by a factor of up to 25.…”

Mathematical Modeling and Numerical Optimization of Composite Structures

“…The solution of these equations can be obtained using the methods of solving boundary-value problems for systems of ordinary differential equations. For that purpose, the modified collocation and least-residuals method [19][20][21] were applied.…”

Solving the biharmonic equation with high order accuracy in irregular domains by the least squares collocation method

“…В настоящей статье делается акцент на решении с повышенной точностью краевых задач для неоднородного бигармонического уравнения в нерегулярных областях методом коллокации и наименьших квадратов (КНК) [17,[19][20][21][22][23]. Актуальность данного направления очевидна, поскольку многие явления в природе, которые моделируются с помощью численных методов, происходят в областях со сложной геометрической формой.…”

“…Проекционно-сеточный метод КНК [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] возник относительно недавно и сочетает в себе свойства метода коллокации и метода наименьших квадратов (МНК). В методе КНК путем проектирования задачи для PDE в конечномерное линейное функциональное пространство ставится в соответствие приближенная задача, решение которой сводится к решению системы линейных алгебраических уравнений (СЛАУ).…”