Research output: Contribution to journal › Article › peer-review
Solving elliptic equations in polygonal domains by the least squares collocation method. / Shapeev, V. P.; Bryndin, L. S.; Belyaev, V. A.
In: Вестник ЮУрГУ. Серия "Математическое моделирование и программирование", Vol. 12, No. 3, 08.2019, p. 140-152.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Solving elliptic equations in polygonal domains by the least squares collocation method
AU - Shapeev, V. P.
AU - Bryndin, L. S.
AU - Belyaev, V. A.
PY - 2019/8
Y1 - 2019/8
N2 - The paper considers a new version of the least squares collocation (LSC) method for the numerical solution of boundary value problems for elliptic equations in polygonal domains, in particular, in multiply connected domains. The implementation of this approach and numerical experiments are performed on the examples of the inhomogeneous biharmonic and Poisson equations. As an application, we use the nonhomogeneous biharmonic equation to simulate the stress-strain state of isotropic elastic thin plate of polygonal form under the action of transverse load. The new version of the LSC method is based on the triangulation of the original domain. Therefore, this approach is fundamentally different from the previous more complicated versions of the LSC method proposed to solve the boundary value problems for partial derivative equations in irregular domains. We make the numerical experiments on the convergence of the approximate solution to various problems on a sequence of grids. The experiments show that the solution to the problems converges with high order and, in the case of the known analytical solution, matches with high accuracy with the analytical solution to the test problems.
AB - The paper considers a new version of the least squares collocation (LSC) method for the numerical solution of boundary value problems for elliptic equations in polygonal domains, in particular, in multiply connected domains. The implementation of this approach and numerical experiments are performed on the examples of the inhomogeneous biharmonic and Poisson equations. As an application, we use the nonhomogeneous biharmonic equation to simulate the stress-strain state of isotropic elastic thin plate of polygonal form under the action of transverse load. The new version of the LSC method is based on the triangulation of the original domain. Therefore, this approach is fundamentally different from the previous more complicated versions of the LSC method proposed to solve the boundary value problems for partial derivative equations in irregular domains. We make the numerical experiments on the convergence of the approximate solution to various problems on a sequence of grids. The experiments show that the solution to the problems converges with high order and, in the case of the known analytical solution, matches with high accuracy with the analytical solution to the test problems.
KW - Least squares collocation method
KW - Nonhomogeneous biharmonic equation
KW - Poisson’s equation
KW - Polygonal multiply connected domain
KW - Stress-strain state
KW - least squares collocation method
KW - polygonal multiply connected domain
KW - Poisson's equation
KW - nonhomogeneous biharmonic equation
KW - stress-strain state
UR - http://www.scopus.com/inward/record.url?scp=85072205707&partnerID=8YFLogxK
U2 - 10.14529/mmp190312
DO - 10.14529/mmp190312
M3 - Article
AN - SCOPUS:85072205707
VL - 12
SP - 140
EP - 152
JO - Вестник ЮУрГУ. Серия "Математическое моделирование и программирование"
JF - Вестник ЮУрГУ. Серия "Математическое моделирование и программирование"
SN - 2071-0216
IS - 3
ER -
ID: 21541802