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G.i. Shishkin - One of the best experts on this subject based on the ideXlab platform.
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Grid Approximation of a singularly perturbed parabolic reaction-diffusion equation with a fast-moving source
Computational Mathematics and Mathematical Physics, 2020Co-Authors: G.i. ShishkinAbstract:The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a powerful fast-moving source (distributed over space but close in form to a pulse) is considered on an interval. The highest derivative of the equation is multiplied by a parameter e. The typical duration of action of the moving source (as it passes through the points of the interval) is described by a parameter λ. The parameters e and λ take arbitrary values in the half-open interval (0, 1]. The solution to the problem has singularities such as boundary and transition layers. For small values of λ, a power singularity generated by the source passing near the boundary arises in a neighborhood of the region where the boundary and transition layers interact. Difference schemes based on classical Grid Approximations of the boundary value problem on rectangular meshes are examined. It is shown that there are no schemes in this class that converge λ-uniformly. Condensing meshes are used to construct schemes that converge at a rate of O(λ -ν (N -2 ln 2 N + N -1 0 )), i.e., e-uniformly and almost λ-uniformly, where N and No determine the numbers of mesh points in x and t and ν > 0 is an arbitrary sufficiently small number.
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Grid Approximation of the solution to the Blasius equation and of its derivatives
Computational Mathematics and Mathematical Physics, 2020Co-Authors: G.i. ShishkinAbstract:A boundary value problem for the Blasius equation in the boundary-layer theory is considered on a half-line. The Blasius equation represents a third-order quasilinear equation in which the coefficient (desired function) of the second-order derivative increases at infinity. For the boundary value problem, formal and constructive difference schemes are constructed on uniform meshes with infinite and finite numbers of mesh points, respectively. The difference schemes approximate (on the half-line) the solution to the boundary value problem and its derivatives involved in the differential equation on the half-line. In the construction and substantiation of the difference schemes, the original boundary value problem is reduced to an equivalent boundary value problem for a system of two equations. The solutions to the equivalent problem (and its Grid Approximations) are analyzed by using majorant functions. Conditions are presented under which the difference solutions and their difference derivatives (up to the third order) converge with a close-to-first order as N → ∞, where N is the number of mesh points in a constructive scheme (or the number of mesh points per unit length in a formal scheme).
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Grid Approximation of a parabolic convection-diffusion equation on a priori adapted Grids: ε-uniformly convergent schemes
Computational Mathematics and Mathematical Physics, 2020Co-Authors: G.i. ShishkinAbstract:The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted Grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted Grids is based on a majorant of the singular component of the Grid solution that makes it possible to a priori find a subdo- main in which the Grid solution should be further refined given the perturbation parameter e , the size of the uniform mesh in x , the desired accuracy of the Grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the Grid problems are solved on uniform Grids. The error of the solution thus constructed weakly depends on e . The scheme converges almost e -uniformly; namely, it converges under the condition N -1 = o ( e ν ) , where ν = ν ( K ) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform Grid is used instead of a uniform one at the final K th iteration, the difference scheme converges e -uniformly. For this piecewise uniform Grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known e -uniformly convergent schemes on piecewise uniform Grids. DOI: 10.1134/S0965542508060080
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Grid Approximation of a parabolic convection diffusion equation on a priori adapted Grids e uniformly convergent schemes
Computational Mathematics and Mathematical Physics, 2008Co-Authors: G.i. ShishkinAbstract:The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted Grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted Grids is based on a majorant of the singular component of the Grid solution that makes it possible to a priori find a subdo- main in which the Grid solution should be further refined given the perturbation parameter e , the size of the uniform mesh in x , the desired accuracy of the Grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the Grid problems are solved on uniform Grids. The error of the solution thus constructed weakly depends on e . The scheme converges almost e -uniformly; namely, it converges under the condition N -1 = o ( e ν ) , where ν = ν ( K ) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform Grid is used instead of a uniform one at the final K th iteration, the difference scheme converges e -uniformly. For this piecewise uniform Grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known e -uniformly convergent schemes on piecewise uniform Grids. DOI: 10.1134/S0965542508060080
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Grid Approximation of singularly perturbed parabolic equations with piecewise continuous initial boundary conditions
Proceedings of the Steklov Institute of Mathematics, 2007Co-Authors: G.i. ShishkinAbstract:The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter ɛ2, ɛ e (0, 1]. For small values of the parameter ɛ, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width ɛ), respectively, which have bounded smoothness for fixed values of the parameter ɛ. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing Grids (piecewise uniform Grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge ɛ-uniformly with the second order of accuracy in x and the first order of accuracy in t, up to logarithmic factors.
Grigorii I Shishkin - One of the best experts on this subject based on the ideXlab platform.
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Grid Approximation of a singularly perturbed parabolic reaction diffusion equation on a ball
International Conference on Numerical Analysis and Its Applications, 2009Co-Authors: L P Shishkina, Grigorii I ShishkinAbstract:We consider a Dirichlet problem on a ball for a singularly perturbed parabolic reaction-diffusion equation. The Laplacian in the equation is multiplied by a perturbation parameter e 2, where e ∈ (0,1]. The solution of such a problem exhibits the parabolic boundary layer in a neighbourhood of the ball boundary as e→0. Using the integro-interpolational method and the condensing mesh method, we construct conservative finite difference schemes whose solutions converge e-uniformly.
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NAA - Grid Approximation of a Singularly Perturbed Parabolic Reaction-Diffusion Equation on a Ball
Lecture Notes in Computer Science, 2009Co-Authors: L P Shishkina, Grigorii I ShishkinAbstract:We consider a Dirichlet problem on a ball for a singularly perturbed parabolic reaction-diffusion equation. The Laplacian in the equation is multiplied by a perturbation parameter e 2, where e ∈ (0,1]. The solution of such a problem exhibits the parabolic boundary layer in a neighbourhood of the ball boundary as e→0. Using the integro-interpolational method and the condensing mesh method, we construct conservative finite difference schemes whose solutions converge e-uniformly.
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Grid Approximation of singularly perturbed parabolic equations with moving boundary layers
Mathematical Modelling and Analysis, 2008Co-Authors: Grigorii I ShishkinAbstract:Abstract A Grid Approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation in a domain with boundaries moving along the x‐axis in the positive direction. For small values of the parameter ϵ (that is the coefficient of the highest‐order derivative in the equation, ϵ ∈ (0,1]), a moving boundary layer appears in a neighbourhood of the left lateral boundary SL 1. It turns out that, in the class of difference schemes on rectangular Grids condensing in a neighbourhood of SL 1 with respect to x and t, there do not exist schemes that converge even under the condition P 0 −1 Â ϵ1/2, where P 0 is the total number of nodes in the meshes used, that is, P 0 Â N N 0, where the values N and N 0 define the numbers of mesh points in x and t. On such meshes, convergence under the condition N −1 + N 0 −1 ≤ ϵ1/4 cannot be achieved. Examination of widths similar to Kolmogorov's widths allows us to establish necessary and sufficient conditions for the ϵ‐uniform conv...
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Grid Approximation of singularly perturbed parabolic reaction diffusion equations with piecewise smooth initial boundary conditions
Mathematical Modelling and Analysis, 2007Co-Authors: Grigorii I ShishkinAbstract:Abstract A Dirichlet problem is considered for a singularly perturbed parabolic reaction–diffusion equation with piecewise smooth initial‐boundary conditions on a rectangular domain. The higher‐order derivative in the equation is multiplied by a parameter ϵ 2; ϵ ? (0, 1]. For small values of ϵ, a boundary and an interior layer arises, respectively, in a neighbourhood of the lateral part of the boundary and in a neighbourhood of the characteristic of the reduced equation passing through the point of nonsmoothness of the initial function. Using the method of special Grids condensing either in a neighbourhood of the boundary layer or in neighbourhoods of the boundary and interior layers, special ϵ‐uniformly convergent difference schemes are constructed and investigated. It is shown that the convergence rate of the schemes crucially depends on the type of nonsmoothness in the initial–boundary conditions.
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Grid Approximation of singularly perturbed parabolic reaction‐diffusion equations with piecewise smooth initial‐boundary conditions
Mathematical Modelling and Analysis, 2007Co-Authors: Grigorii I ShishkinAbstract:Abstract A Dirichlet problem is considered for a singularly perturbed parabolic reaction–diffusion equation with piecewise smooth initial‐boundary conditions on a rectangular domain. The higher‐order derivative in the equation is multiplied by a parameter ϵ 2; ϵ ? (0, 1]. For small values of ϵ, a boundary and an interior layer arises, respectively, in a neighbourhood of the lateral part of the boundary and in a neighbourhood of the characteristic of the reduced equation passing through the point of nonsmoothness of the initial function. Using the method of special Grids condensing either in a neighbourhood of the boundary layer or in neighbourhoods of the boundary and interior layers, special ϵ‐uniformly convergent difference schemes are constructed and investigated. It is shown that the convergence rate of the schemes crucially depends on the type of nonsmoothness in the initial–boundary conditions.
Emine Celiker - One of the best experts on this subject based on the ideXlab platform.
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A fourth order block-hexagonal Grid Approximation for the solution of Laplace’s equation with singularities
Advances in Difference Equations, 2015Co-Authors: A A Dosiyev, Emine CelikerAbstract:The hexagonal Grid version of the block-Grid method, which is a difference-analytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are from the Holder classes \(C^{4,\lambda}\), \(0
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a fourth order block hexagonal Grid Approximation for the solution of laplace s equation with singularities
Advances in Difference Equations, 2015Co-Authors: A A Dosiyev, Emine CelikerAbstract:The hexagonal Grid version of the block-Grid method, which is a difference-analytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are from the Holder classes \(C^{4,\lambda}\), \(0<\lambda<1\), the uniform error is of order \(O(h^{4})\), h is the step size, when the hexagonal Grid is applied in the ‘nonsingular’ part of the domain. Moreover, in each of the finite neighborhoods of the singular corners (‘singular’ parts), the approximate solution is defined as a quadrature Approximation of the integral representation of the harmonic function, and the errors of any order derivatives are estimated. Numerical results are presented in order to demonstrate the theoretical results obtained.
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Approximation on the hexagonal Grid of the dirichlet problem for laplace s equation
Boundary Value Problems, 2014Co-Authors: A A Dosiyev, Emine CelikerAbstract:The fourth order matching operator on the hexagonal Grid is constructed. Its application to the interpolation problem of the numerical solution obtained by hexagonal Grid Approximation of Laplace’s equation on a rectangular domain is investigated. Furthermore, the constructed matching operator is applied to justify a hexagonal version of the combined Block-Grid method for the Dirichlet problem with corner singularity. Numerical examples are illustrated to support the analysis made.
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Approximation on the hexagonal Grid of the Dirichlet problem for Laplace’s equation
Boundary Value Problems, 2014Co-Authors: A A Dosiyev, Emine CelikerAbstract:The fourth order matching operator on the hexagonal Grid is constructed. Its application to the interpolation problem of the numerical solution obtained by hexagonal Grid Approximation of Laplace’s equation on a rectangular domain is investigated. Furthermore, the constructed matching operator is applied to justify a hexagonal version of the combined Block-Grid method for the Dirichlet problem with corner singularity. Numerical examples are illustrated to support the analysis made.
Eric T. Chung - One of the best experts on this subject based on the ideXlab platform.
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Generalized Multiscale Finite Element method for multicontinua unsaturated flow problems in fractured porous media
Journal of Computational and Applied Mathematics, 2020Co-Authors: Denis Spiridonov, Maria Vasilyeva, Eric T. ChungAbstract:Abstract In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that is coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that is generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine Grid Approximation based on the finite element method and Discrete Fracture Model (DFM) approach. In this model, we construct an unstructured fine Grid that takes into account complex fracture geometries for two and three dimensional formulations. Due to construction of the unstructured Grid, the fine Grid Approximation leads to the very large system of equations. For reduction of the discrete system size, we develop a multiscale method for coarse Grid Approximation of the coupled problem using Generalized Multiscale Finite Element Method (GMsFEM). In this method, we construct coupled multiscale basis functions that are used to construct highly accurate coarse Grid Approximation. The multiscale method allowed us to capture detailed interactions between multiple continua. The adaptive approach is investigated, where we consider two approaches for multiscale basis functions construction: (1) based on the spectral characteristics of the local problems and (2) using simplified multiscale basis functions. We investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations. We present a comparison of the relative error for different number of basis functions and for adaptive approach. Numerical results illustrate that the presented method provides accurate solution of the unsaturated multicontinua problem on the coarse Grid with huge reduction of the discrete system size.
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Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media
Journal of Computational and Applied Mathematics, 2020Co-Authors: Aleksei Tyrylgin, Maria Vasilyeva, Denis Spiridonov, Eric T. ChungAbstract:Abstract In this paper, we consider a poroelasticity problem in heterogeneous multicontinuum media that is widely used in simulations of the unconventional hydrocarbon reservoirs and geothermal fields. Mathematical model contains a coupled system of equations for pressures in each continuum and effective equation for displacement with volume force sources that are proportional to the sum of the pressure gradients for each continuum. To illustrate the idea of our approach, we consider a dual continuum background model with discrete fracture networks that can be generalized to a multicontinuum model for poroelasticity problem in complex heterogeneous media. We present a fine Grid Approximation based on the finite element method and Discrete Fracture Model (DFM) approach for two and three-dimensional formulations. The coarse Grid Approximation is constructed using the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions for displacement and pressures in multicontinuum media. We present numerical results for the two and three dimensional model problems in heterogeneous fractured porous media. We investigate relative errors between reference fine Grid solution and presented coarse Grid Approximation using GMsFEM with different numbers of multiscale basis functions. Our results indicate that the proposed method is able to give accurate solutions with few degrees of freedoms.
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FDM - Upscaled Model for Mixed Dimensional Coupled Flow Problem in Fractured Porous Media Using Non-local Multicontinuum (NLMC) Method
Finite Difference Methods. Theory and Applications, 2020Co-Authors: Maria Vasilyeva, Eric T. Chung, Tat Leung, Yalchin Efendiev, Yating WangAbstract:In this paper, we consider a mixed dimensional discrete fracture model with highly conductive fractures. Mathematically the problem is described by a coupled system of equations consisting a d - dimensional equation for flow in porous matrix and a \((d-1)\) - dimensional equation for fracture networks with a specific exchange term for coupling them. For the numerical solution on the fine Grid, we construct unstructured mesh that is conforming with fracture surface and use the finite element Approximation. Fine Grid Approximation typically leads to very large systems of equations since it resolves the fracture networks, and therefore some multiscale methods or upscaling methods should be applied. The main contribution of this paper is that we propose a new upscaled model using Non-local multi-continuum (NLMC) method and construct an effective coarse Grid Approximation. The upscaled model has only one additional coarse degree of freedom (DOF) for each fracture network. We will present results of the numerical simulations using our proposed upscaling method to illustrate its performance.
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Constrained energy minimization based upscaling for coupled flow and mechanics
Journal of Computational Physics, 2020Co-Authors: Maria Vasilyeva, Eric T. Chung, Yalchin EfendievAbstract:Abstract In this paper, our aim is to present (1) an embedded fracture model (EFM) for coupled flow and mechanics problem based on the dual continuum approach on the fine Grid and (2) an upscaled model for the resulting fine Grid equations. The mathematical model is described by the coupled system of equation for displacement, fracture and matrix pressures. For a fine Grid Approximation, we use the finite volume method for flow problem and finite element method for mechanics. Due to the complexity of fractures, solutions have a variety of scales, and fine Grid Approximation results in a large discrete system. Our second focus in the construction of the upscaled coarse Grid poroelasticity model for fractured media. Our upscaled approach is based on the nonlocal multicontinuum (NLMC) upscaling for coupled flow and mechanics problem, which involves computations of local basis functions via an energy minimization principle. This concept allows a systematic upscaling for processes in the fractured porous media, and provides an effective coarse scale model whose degrees of freedoms have physical meaning. We obtain a fast and accurate solver for the poroelasticity problem on a coarse Grid and, at the same time, derive a novel upscaled model. We present numerical results for the two dimensional model problem.
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Upscaling of the single-phase flow and heat transport in fractured geothermal reservoirs using nonlocal multicontinuum method
Computational Geosciences, 2019Co-Authors: Maria Vasilyeva, Masoud Babaei, Eric T. Chung, Valentin AlekseevAbstract:In this work, we consider a single-phase flow and heat transfer problem in fractured geothermal reservoirs. Mixed dimensional problems are considered, where the temperature and pressure equations are solved for porous matrix and fracture networks with transfer term between them. For the fine-Grid Approximation, a finite volume method with embedded fracture model is employed. To reduce size of the fine-Grid system, an upscaled coarse-Grid model is constructed using the nonlocal multicontinuum (NLMC) method. We present numerical results for two-dimensional problems with complex fracture distributions and investigate an accuracy of the proposed method. The simulations using upscaled model provide very accurate solutions with significant reduction in the dimension of problem.
A A Dosiyev - One of the best experts on this subject based on the ideXlab platform.
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A fourth order block-hexagonal Grid Approximation for the solution of Laplace’s equation with singularities
Advances in Difference Equations, 2015Co-Authors: A A Dosiyev, Emine CelikerAbstract:The hexagonal Grid version of the block-Grid method, which is a difference-analytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are from the Holder classes \(C^{4,\lambda}\), \(0
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a fourth order block hexagonal Grid Approximation for the solution of laplace s equation with singularities
Advances in Difference Equations, 2015Co-Authors: A A Dosiyev, Emine CelikerAbstract:The hexagonal Grid version of the block-Grid method, which is a difference-analytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are from the Holder classes \(C^{4,\lambda}\), \(0<\lambda<1\), the uniform error is of order \(O(h^{4})\), h is the step size, when the hexagonal Grid is applied in the ‘nonsingular’ part of the domain. Moreover, in each of the finite neighborhoods of the singular corners (‘singular’ parts), the approximate solution is defined as a quadrature Approximation of the integral representation of the harmonic function, and the errors of any order derivatives are estimated. Numerical results are presented in order to demonstrate the theoretical results obtained.
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Approximation on the hexagonal Grid of the dirichlet problem for laplace s equation
Boundary Value Problems, 2014Co-Authors: A A Dosiyev, Emine CelikerAbstract:The fourth order matching operator on the hexagonal Grid is constructed. Its application to the interpolation problem of the numerical solution obtained by hexagonal Grid Approximation of Laplace’s equation on a rectangular domain is investigated. Furthermore, the constructed matching operator is applied to justify a hexagonal version of the combined Block-Grid method for the Dirichlet problem with corner singularity. Numerical examples are illustrated to support the analysis made.
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Approximation on the hexagonal Grid of the Dirichlet problem for Laplace’s equation
Boundary Value Problems, 2014Co-Authors: A A Dosiyev, Emine CelikerAbstract:The fourth order matching operator on the hexagonal Grid is constructed. Its application to the interpolation problem of the numerical solution obtained by hexagonal Grid Approximation of Laplace’s equation on a rectangular domain is investigated. Furthermore, the constructed matching operator is applied to justify a hexagonal version of the combined Block-Grid method for the Dirichlet problem with corner singularity. Numerical examples are illustrated to support the analysis made.
