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A posteriori stopping criteria for space-time domain decomposition for the heat equation in mixed formulations

Sarah Ali Hassan, Caroline Japhet, Martin Vohralík

2018
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Electronic Transactions on Numerical Analysis
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We propose and analyse a posteriori estimates for global-in-time, nonoverlapping domain decomposition methods for heterogeneous and anisotropic porous media diffusion problems. We consider mixed formulations, with a lowest-order Raviart-Thomas-Nédélec discretization, often used for such problems. Optimized Robin transmission conditions are employed on the space-time interface between subdomains, and different time grids are used to adapt to different time scales in the subdomains. Our
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... ains. Our estimators allow to distinguish the spatial discretization, the temporal discretization, and the domain decomposition error components. We design an adaptive space-time domain decomposition algorithm, wherein the iterations are stopped when the domain decomposition error does not affect significantly the global error. Thus, a guaranteed bound on the overall error is obtained on each iteration of the space-time domain decomposition algorithm, and simultaneously important savings in terms of the number of domain decomposition iterations can be achieved. Numerical results for two-dimensional problems with strong heterogeneities and local time stepping are presented to illustrate the performance of our adaptive domain decomposition algorithm. A posteriori stopping criteria for space-time domain decomposition for the heat equation in mixed formulations2 is the Dirichlet boundary condition, and p 0 ∈ H 1 (Ω) is the initial condition with p 0 | Γ D = g D (·, 0)| Γ D . Furthermore, f ∈ L 2 (Ω × (0, T )) is the source term, n is the outward unit normal vector to ∂Ω, and S S S is a symmetric, bounded, and uniformly positive definite tensor whose terms are for simplicity supposed piecewise constant on the mesh T h of Ω defined below and constant in time. We consider global-in-time optimized Schwarz method which uses the optimized Schwarz waveform relaxation (OSWR) approach [31, 49] . This is an iterative method that computes in the subdomains over the whole space-time interval, exchanging space-time boundary data through transmission conditions on the space-time interfaces. The OSWR algorithm uses more general (Robin or Ventcell) transmission operators in which coefficients can be optimized to improve convergence rates, see [31, 42, 49] . The optimization of the Robin (or Ventcell) parameters was analyzed in [10, 12] . Generalizations to heterogeneous problems with nonmatching time grids were introduced in [11, 13, 30, 34, 35, 36, 37, 41, 40, 39] . More precisely, in [13, 36, 37] , a discontinuous Galerkin (DG) method for the time discretization of the OSWR algorithm was introduced and analyzed for the case of nonconforming time grids. A suitable time projection between subdomains is defined using an optimal projection algorithm as in [33, 32] with no additional grid. In the context of mixed finite elements, which are mass conservative and handle well heterogeneous and anisotropic diffusion tensors, we refer also to [23, 41, 39] . The multi-domain problem can actually be reformulated as an interface problem (see [21] , [39], or [3]) that can be solved by various iterative methods, such as block-Jacobi or GMRES. Our first objective in this contribution is to design a posteriori estimates valid on each step of the space-time domain decomposition algorithm. For general algebraic iterative solvers, several techniques with residual-based estimates have been developed, see [9, 6, 7] , see also [53, 50, 57] for goal-oriented a posteriori error estimates. A general framework for any numerical method and any algebraic solver has been introduced in [26], building on the ideas from [43], and has been extended to coupled unsteady nonlinear and degenerate problems in [15, 20] . For lowest-order time discretizations, this approach is based on a H 1 (Ω)conforming reconstruction of the potential, continuous and piecewise affine in time, and an equilibrated H(div, Ω)-conforming reconstruction of the flux, piecewise constant in time. It yields a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative (see [61, 25] ), without unknown constants. Using a globally equivalent norm, which contains also the temporal jumps of the numerical solution, it leads to local space-time efficiency, see the recent contribution [24] . Recently, a posteriori error estimates and stopping criteria for non-overlapping domain decomposition algorithms such as FETI [28] or BDD [48, 17] , have been proposed in [58, 59] . Both upper and lower bounds for the overall error are derived, and the discretization and the domain decomposition error components are distinguished. Also this approach is based on H 1 (Ω)-conforming potential and H(div, Ω)-conforming flux reconstructions, and follows the a posteriori techniques of of [55, 46, 56, 27] . A key observation is that such reconstructions can be easily obtained when the solution approach involves subdomain problems with both Dirichlet and Neumann interface conditions on each domain decomposition (DD) iteration, as this is the case for FETI or BDD. For domain decomposition strategies with more general interface conditions, and where neither the conformity of the flux nor that of the potential is preserved (as long as the convergence is not reached), a new adaptive domain decomposition algorithm has been introduced in [3]. More precisely, three reconstructions are proposed: a flux reconstruction that is globally H(div, Ω)-conforming and locally conservative in each mesh element, based on the construction of [54, Section 3.5.2], as well as two H 1 -conforming potential reconstructions, one globally on Ω, relying on the averaging operator I av , see [1, 44, 14] , and another on each subdomain Ω i , which introduces weights on the interfaces and whose goal is to separate the DD and the discretization components. Then, error control is achieved on each step and an adaptive domain decomposition algorithm is proposed wherein the iterations are stopped when the domain decomposition error does not affect significantly the overall error. This paper is a continuation of [3]: we provide a new approach that makes it possible to extend this adaptive domain decomposition algorithm to model coupled time-dependent diffusion problems. We focus on mixed finite element discretizations in the subdomains and extend the approaches from [45, 62, 2, 63, 54, 25, 24, 3] for a posteriori error estimates. We first build a flux reconstruction that is globally H(div, Ω)conforming, locally conservative in each mesh element, and piecewise constant in time. Following [3], a

doi:10.1553/etna_vol49s151
fatcat:aebiliojyjcb5dzlojmvrj4yiq