Geodesic
Solving the biharmonic equation with high order accuracy in irregular domains by the least squares collocation method
Numerical methods and programming, Tome 19 (2018) no. 4, pp. 340-355.

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This paper addresses a new version of the least squares collocation (LSC) method of high order accuracy proposed and implemented for the numerical solution of the nonhomogeneous biharmonic equation. The differential problem is projected onto a polynomial space of fourth and eighth degrees by the LSC method. The algorithm implemented is applied in irregular domains. The boundaries of these domains are given by analytical curves, in particular, by splines. The irregular domain is embedded in a rectangle covered by a regular grid with rectangular cells. In this paper we use the irregular cells (i-cells) which are cut off by the domain boundary from the rectangular cells of the initial regular grid. All i-cells are divided into two classes: the independent and dependent ones. The center of a dependent cell is located outside the domain by contrast with the center of an independent cell. The idea of attaching elongated dependent irregular cells to the neighboring ones is used. A separate piece of the analytical solution is constructed in the combined cells. The collocation and matching points located outside the domain are used to approximate the differential equations in the boundary cells. These two approaches allow us to essentially reduce the conditionality of the corresponding system of linear algebraic equations. It is shown that the approximate solutions obtained by the LSC method converge with an increased order and coincide with the analytical solutions of the test problems with high accuracy in the case of the known solution. The numerical results are compared with those found by other authors who used a high order finite difference method. The nonhomogeneous biharmonic equation is used to model the stress-strain state of isotropic thin irregular plates as an application.
Keywords: least squares collocation method, nonhomogeneous biharmonic equations, high order approximation, irregular domains, double splines. 
@article{VMP_2018_19_4_a2,
     author = {V. P. Shapeev and V. A. Belyaev},
     title = {Solving the biharmonic equation with high order accuracy in irregular domains by the least squares collocation method},
     journal = {Numerical methods and programming},
     pages = {340--355},
     publisher = {mathdoc},
     volume = {19},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2018_19_4_a2/}
}
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V. P. Shapeev; V. A. Belyaev. Solving the biharmonic equation with high order accuracy in irregular domains by the least squares collocation method. Numerical methods and programming, Tome 19 (2018) no. 4, pp. 340-355. http://geodesic.mathdoc.fr/item/VMP_2018_19_4_a2/