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Numerical solution of an elliptic equation with boundary layers for a half-strip

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Title: Numerical solution of an elliptic equation with boundary layers for a half-strip
Authors: Zadorin, A. I.
Publisher Information: Russian Academy of Sciences - RAS (Rossiĭskaya Akademiya Nauk - RAN), Siberian Branch (Sibirskoe Otdelenie), Institute of Computational Technologies (Institut Vychislitel'nykh Tekhnologiĭ), Novosibirsk
Subject Terms: Finite difference methods for boundary value problems involving PDEs, domain truncation, Error bounds for boundary value problems involving PDEs, \(\varepsilon \)-uniform convergence, elliptic equation, finite-difference method, small parameter, Degenerate elliptic equations, nonuniform mesh, Stability and convergence of numerical methods for boundary value problems involving PDEs, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, singular perturbation, Singular perturbations in context of PDEs
Description: A boundary value problem for an elliptic equation \[ \varepsilon u_{xx}+\varepsilon u_{yy}-a(x,y)u_x-b(x,y)u=f(x,y) \tag{1} \] with a small parameter \(\varepsilon\) at the highest derivatives is considered in a semi-infinite half-strip \(D=\left\{0 \leq x \leq \infty , 0 \leq y \leq 1 \right\}\). At infinity, the following condition is imposed: \[ \lim_{x\to \infty}u(x,y)=0. \tag{2} \] The domain \(D\) is replaced by the bounded rectangular region \(D_L=\left\{0 \leq x \leq L, \;0 \leq y \leq 1 \right \}\) and, on the boundary \( x = L \), a condition is imposed asserting that equation (1) degenerates on the coordinate \( x \): \[ a(x,y)u_x+b(x,y)u-\varepsilon u_{yy}\Big {|}_{x=L}=-f(x,y). \tag{3} \] It is proven that such a translation of the boundary condition from infinity preserves the boundary condition (2) as \(L \rightarrow \infty\). Estimates for the replacement error are obtained. For the problem in the rectangular domain \(D_L\), the difference scheme is constructed on the special nonuniform grid condensing near the boundaries \( y = 0 \) and \( y = 1 \), where the solution can possess boundary layers. It is proven that the finite-difference method under consideration is \(\varepsilon\)-uniform. This part of the article is somewhat related to the article by \textit{G. I. Shishkin} [U.S.S.R. Comput. Math. Math. Phys. 26, No. 4, 38-46 (1986; Zbl 0635.65110); translation from Zh. Vychisl. Mat. Mat. Fiz. 26, No. 7, 1019-1031 (1986; Zbl 0622.65078)]. Numerical results for the equation (1) with \(a = 1, \;b =2 \) are presented.
Document Type: Article
File Description: application/xml
Access URL: https://zbmath.org/1432195
Accession Number: edsair.c2b0b933574d..3eec473b6c5573fc5209c5215ef1ffdd
Database: OpenAIRE
Description
Abstract:A boundary value problem for an elliptic equation \[ \varepsilon u_{xx}+\varepsilon u_{yy}-a(x,y)u_x-b(x,y)u=f(x,y) \tag{1} \] with a small parameter \(\varepsilon\) at the highest derivatives is considered in a semi-infinite half-strip \(D=\left\{0 \leq x \leq \infty , 0 \leq y \leq 1 \right\}\). At infinity, the following condition is imposed: \[ \lim_{x\to \infty}u(x,y)=0. \tag{2} \] The domain \(D\) is replaced by the bounded rectangular region \(D_L=\left\{0 \leq x \leq L, \;0 \leq y \leq 1 \right \}\) and, on the boundary \( x = L \), a condition is imposed asserting that equation (1) degenerates on the coordinate \( x \): \[ a(x,y)u_x+b(x,y)u-\varepsilon u_{yy}\Big {|}_{x=L}=-f(x,y). \tag{3} \] It is proven that such a translation of the boundary condition from infinity preserves the boundary condition (2) as \(L \rightarrow \infty\). Estimates for the replacement error are obtained. For the problem in the rectangular domain \(D_L\), the difference scheme is constructed on the special nonuniform grid condensing near the boundaries \( y = 0 \) and \( y = 1 \), where the solution can possess boundary layers. It is proven that the finite-difference method under consideration is \(\varepsilon\)-uniform. This part of the article is somewhat related to the article by \textit{G. I. Shishkin} [U.S.S.R. Comput. Math. Math. Phys. 26, No. 4, 38-46 (1986; Zbl 0635.65110); translation from Zh. Vychisl. Mat. Mat. Fiz. 26, No. 7, 1019-1031 (1986; Zbl 0622.65078)]. Numerical results for the equation (1) with \(a = 1, \;b =2 \) are presented.