In this paper, we consider the mixed Stokes-Darcy problem which describes a fluid flow coupled with a porous media. We present a modified two-grid method for decoupling this mixed model. Stability is proved and optima...
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In this paper, we consider the mixed Stokes-Darcy problem which describes a fluid flow coupled with a porous media. We present a modified two-grid method for decoupling this mixed model. Stability is proved and optimal error estimates are derived. The numerical results show that the modified two-grid method is effective and has the same accuracy as the coupling scheme when we choose h = H-2. (C) 2014 Elsevier B.V. All rights reserved.
A boundary value problem for a second-order nonlinear singular perturbation ordinary differential equation is considered. A method based on Newton and Picard linearizations using a modified Samarskii scheme on a Shish...
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A boundary value problem for a second-order nonlinear singular perturbation ordinary differential equation is considered. A method based on Newton and Picard linearizations using a modified Samarskii scheme on a Shishkin grid for a linear problem is proposed. It is proved that the difference schemes are of second-order and uniformly convergent. To decrease the number of arithmetic operations, a two-grid method is proposed. The results of some numerical experiments are discussed.
The problem of exact controllability of elastic strings has been extensively studied during the last years. We consider the problem of computing numerically the boundary control for a finite-dimensional system obtaine...
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The problem of exact controllability of elastic strings has been extensively studied during the last years. We consider the problem of computing numerically the boundary control for a finite-dimensional system obtained by discretizing in space the 1-d wave equation. More precisely, we analyze whether the controls of numerical approximation schemes converge to the control of the continuous wave equation as the mesh size tends to zero. It is by now well known that, due to high-frequency spurious oscillations, numerical instabilities occur and may lead to the failure of convergence of some apparently natural numerical algorithms. In other words, the classical convergence property of numerical schemes does not guarantee a stable and convergent approximation of controls. Several remedies have been proposed in the literature to compensate for this fact: Tychonoff regularization, Fourier filtering, and mixed finite elements. In this paper we prove that the two-grid method proposed by Glowinski in [J. Comput. Phys., 103 ( 1992), pp. 189-221] to numerically approximate the control of the wave equation converges in 1-d. We prove this result in the context of the finite-element space semidiscretization. Our method of proof relies essentially on the particular properties of the Fourier representation of the initial data of the coarse mesh when projected into the. ne one. The explicit representation formula of the solutions shows that the high-frequency components are modulated by some weights that diminish the effect of these spurious components. This fact, combined with discrete multipliers techniques, allows us to prove uniform observability inequalities. Classical arguments then allow proving the uniform boundedness of the controls and passing to the limit as the mesh size tends to zero. In this way we prove the convergence of the controls of the finite-element semidiscrete approximation of the 1-d wave equation with a boundary control on one extreme. It is important to unde
In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree ra parts per thousand yen1 for a class of quasi-linear elliptic problems in Omega aS,...
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In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree ra parts per thousand yen1 for a class of quasi-linear elliptic problems in Omega aS,a"e(2). We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H (1)-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in a"e (d) ,d=2,3 and use it to establish the convergence of the two-grid method for problems in Omega aS,a"e(3).
In this paper, several two-grid algorithms are presented. For nonsymmetric linear systems, we propose a two-grid algorithm by using the information of the adjoint operator. The solution of the original systems is main...
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In this paper, several two-grid algorithms are presented. For nonsymmetric linear systems, we propose a two-grid algorithm by using the information of the adjoint operator. The solution of the original systems is mainly reduced to a solution of symmetric positive definite (SPD) systems. For nonlinear systems, we present a two-grid algorithm based on the modified Newton method. The solution of the original systems on the fine space is reduced to the solution of two small systems on the coarse space and two similar linear systems (with same stiffness matrix) on the fine space. It is shown that the accuracy (L-2 norm) obtained by this algorithm is as same as the optimal accuracy derived by using two full Newton steps. Additionally, for more practically applications, the ideas of these algorithms can be also extended to the multilevel case. Numerical experiments are given for these new algorithms.
We study numerical methods for a coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. A decoupled and linearized two-grid algorithm is proposed. Numerical analysis and experi...
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We study numerical methods for a coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. A decoupled and linearized two-grid algorithm is proposed. Numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm.
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