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Keywords = Rν-generalized solution

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11 pages, 299 KB  
Article
The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity
by Viktor A. Rukavishnikov and Elena I. Rukavishnikova
Mathematics 2023, 11(15), 3272; https://doi.org/10.3390/math11153272 - 25 Jul 2023
Viewed by 1523
Abstract
Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior [...] Read more.
Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior of the coefficients of the equation. Finite element methods with the first order of convergence rate O(h) have been created to find an approximate solution to these problems. We construct a scheme of the weighted finite element method of high degree of accuracy for the boundary value problem with uncoordinated degeneracy of the input data and singularity of the solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact Rν-generalized solution in the weight set W2,ν+β2+21(Ω,δ) is investigated. The estimation of finite element approximation O(h2) is established. Full article
(This article belongs to the Section E: Applied Mathematics)
14 pages, 307 KB  
Article
On the Existence and Uniqueness of an Rν-Generalized Solution to the Stokes Problem with Corner Singularity
by Viktor A. Rukavishnikov and Alexey V. Rukavishnikov
Mathematics 2022, 10(10), 1752; https://doi.org/10.3390/math10101752 - 20 May 2022
Cited by 9 | Viewed by 2021
Abstract
We consider the Stokes problem with the homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary. We define an Rν-generalized solution of the problem in a nonsymmetric variational formulation. Such defined solution allows us to [...] Read more.
We consider the Stokes problem with the homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary. We define an Rν-generalized solution of the problem in a nonsymmetric variational formulation. Such defined solution allows us to construct numerical methods for finding an approximate solution without loss of accuracy. In the paper, the existence and uniqueness of an Rν-generalized solution in weighted sets is proved. Full article
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