A domain decomposition preconditioner for an advection–diffusion problem

Achdou, Yves; Tallec, Patrick Le; Nataf, Frédéric; Vidrascu, Marina

doi:10.1016/s0045-7825(99)00227-3

Cited by 67 publications

(69 citation statements)

References 12 publications

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“…The RobinRobin preconditioned solver further reduces the residual around the 25 th iteration, while the non-preconditioned system shows little improvement. Achdou et al point out in [1] that in convection-dominated flows, the continuous Robin-Robin preconditioned system operator is close to an idempotent (or periodic) operator of order E/2 where E is the number of elements. They argue that this causes GMRES to stagnate for E/2 steps before converging asymptotically 1 .…”

Section: Variable Wind Resultsmentioning

confidence: 99%

“…Our choice of preconditioner is the Robin-Robin preconditioner developed in [1], which extends the popular Neumann-Neumann preconditioner used to accelerate the convergence of interface solvers corresponding to the Poisson equation, [14], [17], [20] to nonsymmetric systems. This technique uses a pseudo-inverse of the locally defined Schur complement operator, with Robin and Neumann boundary conditions applied to elemental interfaces.…”

Section: Interface Solvementioning

confidence: 99%

“…Achdou et al [1] adapted the Neumann-Neumann preconditioner to non-symmetric convection-diffusion systems by choosing interface boundary conditions that reflect the movement of the flow across element boundaries. In particular, Robin boundary conditions are used instead of Neumann conditions at inflow boundaries where w · n < 0; Neumann boundary conditions are still imposed at outflows where w · n > 0.…”

Section: Interface Solvementioning

confidence: 99%

“…This strategy is known as a Robin-Robin preconditioner. It ensures that the symmetric part of the preconditioning operator is positive-definite (see [1] & [20]), since the underlying bilinear form,…”

Section: Interface Solvementioning

confidence: 99%

“…This added term allows the flow to move between elements, and ensures that the preconditioned system matrix is positive definite, as described in [1] and [20]. In practice, the elemental boundary conditions are determined by considering the sign of the convection term at each element boundary.…”

Section: Interface Solvementioning

confidence: 99%

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Fast iterative solver for convection‐diffusion systems with spectral elements

Lott

Elman

2011

Numerical Methods Partial

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Abstract. We introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion equation. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this method in numerical simulations using a spectral element discretization.Key words. Convection-Diffusion, Domain Decomposition, Preconditioning, Spectral Element Method 1. Introduction. Numerical simulation of fluid flow allows for improved prediction and design of natural and engineered systems such as those involving water, oil, or blood. The interplay between inertial and viscous forces in a fluid flow dictates the length scale where energy is transferred, thus determining the resolution required to capture flow information accurately. This resolution requirement poses computational challenges in situations where the convective nature of the flow dominates diffusive effects. In such flows, convection and diffusion occur on disparate scales, causing sharp flow features that require fine numerical grid resolution. This leads to a large system of equations which is often solved using an iterative method. Exacerbating the challenge of solving a large linear system, the discrete convection-diffusion operator is non-symmetric and poorly conditioned. This leads to slow convergence of iterative solvers. In total, as convection dominates the flow the discrete fluid model becomes exceedingly challenging to solve.In recent years, the spectral element method has gained popularity as a technique for numerical simulation of fluids [9], [18]. This is due in part to the method's high-order accuracy, which produces solutions with low dissipation and low dispersion with relatively few degrees of freedom. Also important is the inherent computational efficiency gained through the use of a hierarchical grid structure based on unstructured macro-elements with fine tensor-structured interiors. This structure has enabled the development of efficient multi-level solvers and preconditioners based on Fast Diagonalization and Domain Decomposition [8], [10], [21]. Application of these techniques, however, has been restricted to symmetric systems.One way to apply such methods to non-symmetric systems is through use of timesplitting techniques, which split the system into symmetric and non-symmetric components. For convection-diffusion systems, the standard method for performing steady and unsteady flow simulations with spectral elements is operator integration factor splitting (OIFS) [12], which requires time integration even in steady flow simulations. Using this standard approach, convection and diffusion are treated separately; convection components are tackled explicitly using a sequence of small time steps that satisfy a CFL condition, and diffusive components are treated implicitly with larger t...

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Section: Variable Wind Resultsmentioning

confidence: 99%

Section: Interface Solvementioning

confidence: 99%

Section: Interface Solvementioning

confidence: 99%

Section: Interface Solvementioning

confidence: 99%

Section: Interface Solvementioning

confidence: 99%

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Fast iterative solver for convection‐diffusion systems with spectral elements

Lott

Elman

2011

Numerical Methods Partial

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Balancing Neumann‐Neumann methods for incompressible Stokes equations

Pavarino

Widlund

2001

Comm Pure Appl Math

110

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Balancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant-pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing NeumannNeumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf-sup constants of the discrete problem and of the coarse subproblem. Numerical results show that the method is quite fast; they are also fully consistent with the theory.

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A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

Dolean¹,

Lanteri²

2001

Numerical Methods in Fluids

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SUMMARYWe report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two-dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo-time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson-type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes.

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A domain decomposition preconditioner for an advection–diffusion problem

Cited by 67 publications

References 12 publications

Fast iterative solver for convection‐diffusion systems with spectral elements

Fast iterative solver for convection‐diffusion systems with spectral elements

Balancing Neumann‐Neumann methods for incompressible Stokes equations

A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

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