Uniform Convergence of the Schwarz Alternating Method for Solving Singularly Perturbed Advection-Diffusion Equations

Mathew, Tarek Poonithara Abraham

doi:10.1137/s0036142995296485

Cited by 34 publications

(25 citation statements)

References 17 publications

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“…A parallel Schwarz iterative method. The linear system (2.4) can be solved by a parallel version of the Schwarz iterative method [26,15,16,35,28,12] which will (under assumptions stated in Section 4) converge geometrically. On a parallel architecture, each processor can in principle be assigned to a different spacetime grid.…”

Section: Global Discretizationmentioning

confidence: 99%

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Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

Mathew¹,

Russo

2002

Math. Comp.

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Abstract. In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain Ω × [0, T ] which is decomposed into an overlapping collection of cylindrical subregions ofT ] are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters h l and τ l . In particular, the different space-time grids need not match on the regions of overlap, and the time steps τ l can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit θ-scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument.Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and heterogeneous hyperbolic-parabolic approximations of parabolic equations.

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Section: Global Discretizationmentioning

confidence: 99%

“…Proof. We follow the construction in Lions [27] (see also [28]). Direct computation of Le −αd l (x) yields…”

Section: Definition a Grid Function Wmentioning

confidence: 99%

Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

Mathew¹,

Russo

2002

Math. Comp.

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“…The Schwarz Alternating Method, of which the method (1.4) is a variant, is used in several cases, especially when the matrix A is nonsymmetric; see, e.g., [8,29], and the references given therein. In these cases, the convergence is studied using max norms and the analytic tool is some maximum principle; see also [26,27].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, our result apply to discretized nonsymmetric equations rather than the continuous case; cf. [29].…”

Section: Introductionmentioning

confidence: 99%

“…The second author thanks Olof B. Widlund for many conversations and explanations on Schwarz methods during many years, in different parts of the world. We thank Tarek P. Mathew for sending us an advance copy of [29], and Michele Benzi for his careful reading of a manuscript and his comments. We also thank Joan-Josep Climent and Carmen Perea for calling our attention to the remark at the end of Sect.…”

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confidence: 99%

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Weighted max norms, splittings, and overlapping additive Schwarz iterations

Frommer

Szyld

1999

Numerische Mathematik

110

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Summary. Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an M -matrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using A-norms when the matrix A is symmetric positive definite. A new theorem concerning P -regular splittings is presented which provides a useful tool for the A-norm bounds. Furthermore, a theory of splittings is developed to represent Algebraic Additive Schwarz Iterations. This representation makes a connection with multisplitting methods. With this representation, and using a comparison theorem, it is shown that a coarse grid correction improves the convergence of Additive Schwarz Iterations when measured in weighted max norm. Classification (1991): 65F10, 65F35, 65M55 Mathematics Subject

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An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations

Roja¹,

Tamilselvan

Geetha

2019

Numerical Methods in Fluids

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In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates. K E Y W O R D Sconvection-diffusion equations, hybrid difference scheme, Schwarz method, singularly perturbed problems, weakly coupled system

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Uniform Convergence of the Schwarz Alternating Method for Solving Singularly Perturbed Advection-Diffusion Equations

Cited by 34 publications

References 17 publications

Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

Weighted max norms, splittings, and overlapping additive Schwarz iterations

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations

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