Extended Variational Formulation for Heterogeneous Partial Differential Equations

Blanco, Pablo; Gervasio, Paola; Quarteroni, Alfio

doi:10.2478/cmam-2011-0008

Cited by 10 publications

(11 citation statements)

References 28 publications

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“…It is also easily adaptable to solve heterogeneous problems since it requires neither an in-depth (specific) knowledge of the differential subproblems nor specific interface conditions as happens for domain decomposition methods with sharp interfaces (see, e.g., [2,4,8]). In particular, in this paper the ICDD method with interface observation is successfully applied to the coupling between advection and advection-diffusion (A-AD) problems.…”

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confidence: 99%

The Interface Control Domain Decomposition (ICDD) Method for Elliptic Problems

Discacciati¹,

Gervasio²,

Quarteroni³

2013

SIAM J. Control Optim.

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Abstract. Interface controls are unknown functions used as Dirichlet or Robin boundary data on the interfaces of an overlapping decomposition designed for solving second order elliptic boundary value problems. The controls are computed through an optimal control problem with either distributed or interface observation. Numerical results show that, when interface observation is considered, the resulting interface control domain decomposition method is robust with respect to coefficients variations; it can exploit nonconforming meshes and provides optimal convergence with respect to the discretization parameters; finally it can be easily used to face heterogeneous advectionadvection-diffusion couplings.Key words. domain decomposition, optimal control, elliptic boundary value problems, nonconforming discretizations, heterogeneous coupling, ICDD AMS subject classifications. 65N55, 49J20, 49K20, 65N30, 65N35 DOI. 10.1137/120890764 1. Introduction. In this paper we propose an overlapping domain decomposition method with interface conditions that are suitable to face both homogenenous and heterogeneous couplings, i.e., with either the same or different operators in the subdomains.We start by considering a family of overlapping domain decomposition methods, originally proposed in [13] and [16], named least square conjugate gradient (LSCG) and virtual control (VC) methods, respectively, designed for second order self-adjoint elliptic problems. These methods reformulate the overlapping multidomain problem as an optimal control problem whose controls are the unknown traces (or fluxes) of the state solutions at the subdomain boundaries. (For this reason we speak about interface controls.) The constraints of the minimization problem are the state equations, the observation is distributed on the overlap, and the controls are found by either minimizing the L 2 or H 1 norm of the jump between state solutions associated with the subdomains that share the same overlap. When the optimal control problem is solved through the optimality system, by following the classical theory of Lions ([15]), we need to solve both the primal and the dual state problems, as well as to compute the jump between the two solutions on the whole overlap. When non-self-adjoint problems are considered, the matrices associated with the discretization of both primal and dual problems have to be built and stored; moreover the same computational grid on the overlap has to be used, in order to avoid heavy interpolation processes from one grid to another. As we will show in [6], the convergence rate of these methods

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mentioning

confidence: 99%

The Interface Control Domain Decomposition (ICDD) Method for Elliptic Problems

Discacciati¹,

Gervasio²,

Quarteroni³

2013

SIAM J. Control Optim.

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“…Under the assumptions introduced for ν and γ , if moreover ∃ β 0 >0 such that

\frac{1}{2} \nabla \cdot boldb + γ \geq β_{0}

in Ω 1 , and under the assumption that λ i ∈Λ i (for i = 1,2) are given, both problems and are well‐posed (see, e.g., ).…”

Section: Interface Control Domain Decomposition For the Coupling Of Amentioning

confidence: 99%

“…In the next section, we write the discrete counterpart of – and prove its well‐posedness. Remark When the computational domain is partitioned into two nonoverlapping subdomains Ω 1 , Ω 2 with sharp interface (i.e., such that

{\overset{normalΩ}{false}}_{1} \cup {\overset{normalΩ}{false}}_{2} = \overset{normalΩ}{false}

, Ω 1 ∩Ω 2 = ∅ and Γ= ∂ Ω 1 ∩ ∂ Ω 2 ), the heterogeneous A–AD coupling has been analyzed in . In , the authors provided a suitable set of interface conditions on the sharp interface Γ of the decomposition, that is,

\begin{matrix} alignleft \\ align-1 \begin{matrix} u_{1} = u_{2} & on {normalΓ}^{in} \\ boldb \cdot boldn u_{1} + ν boldn \cdot \nabla u_{2} - boldb \cdot boldn u_{2} = 0 & on normalΓ \end{matrix} & align-2 \end{matrix}

where n is the unit normal vector to Γ oriented from Ω 1 to Ω 2 .…”

Section: Interface Control Domain Decomposition For the Coupling Of Amentioning

confidence: 99%

“…In such case, a suitable version of the classical Dirichlet–Neumann method can be successfully applied to solve the heterogeneous problem. Other domain decomposition approaches based on virtual controls or extended variational formulation on nonoverlapping subdomains have been addressed in to face similar heterogeneous coupled problems.…”

Section: Interface Control Domain Decomposition For the Coupling Of Amentioning

confidence: 99%

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Interface control domain decomposition methods for heterogeneous problems

Discacciati

Gervasio

Quarteroni

2014

Numerical Methods in Fluids

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SUMMARYThis paper is concerned with the solution of heterogeneous problems by the interface control domain decomposition (ICDD) method, a strategy introduced for the solution of partial differential equations in computational domains partitioned into subdomains that overlap. After reformulating the original boundary value problem by introducing new additional control variables, the unknown traces of the solution at internal subdomain interfaces; the latter are determined by requiring that the (a priori) independent solutions in each subdomain undergo the minimization of a suitable cost functional.We provide an abstract formulation for coupled heterogeneous problems and a general theorem of well‐posedness for the associated ICDD problem. Then, we illustrate and validate an efficient algorithm based on the solution of the Schur‐complement system restricted solely to the interface control variables by considering two kinds of heterogeneous boundary value problems: the coupling between pure advection and advection–diffusion equations and the coupling between Stokes and Darcy equations. In the latter case, we also compare the ICDD method with a classical approach based on the Beavers–Joseph–Saffman conditions. Copyright © 2014 John Wiley & Sons, Ltd.

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“…This prevents us from guaranteeing their coerciveness (see [7]). In this section we present a different approach based on [5,3,4,6] consisting in writing the coupled Darcy-Stokes problem as a system of linear equations on Γ involving both variables λ and η.…”

Section: Augmented Interface Equationsmentioning

confidence: 99%

Augmented Interface Systems for the Darcy-Stokes Problem

Discacciati

2013

Lecture Notes in Computational Science and Engineering

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Summary. In this paper we study interface equations associated to the Darcy-Stokes problem using the classical Steklov-Poincaré approach and a new one called augmented. We compare these two families of methods and characterize at the discrete level suitable preconditioners with additive and multiplicative structures. Finally, we present some numerical results to assess their behavior in presence of small physical parameters.

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Extended Variational Formulation for Heterogeneous Partial Differential Equations

Cited by 10 publications

References 28 publications

The Interface Control Domain Decomposition (ICDD) Method for Elliptic Problems

The Interface Control Domain Decomposition (ICDD) Method for Elliptic Problems

Interface control domain decomposition methods for heterogeneous problems

Augmented Interface Systems for the Darcy-Stokes Problem

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