On a Multiscale Hybrid-Mixed Method for Advective-Reactive Dominated Problems with Heterogeneous Coefficients

Harder, Christopher; Paredes, Diego; Valentin, Frédéric

doi:10.1137/130938499

Cited by 31 publications

(34 citation statements)

References 20 publications

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“…The global problem needs for its construction the solution of local problems that fulfill the role of upscaling the under-mesh structures. Introduced and analysed in [24,2,31] for the Laplace (Darcy) equation, the MHM method has been further extended to other elliptic problems in [25,23] as well as to mixed and hyperbolic models in [3] and [29], respectively. See also [26] for an abstract setting for the MHM method.…”

Section: Introductionmentioning

confidence: 99%

The multiscale hybrid mixed method in general polygonal meshes

et al. 2020

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This work extends the general form of the Multiscale Hybrid-Mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and k + 1, k ≥ 0, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree k + 1 posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order k + 1 and k + 2 in the broken H 1 and L 2 norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides.

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Section: Introductionmentioning

confidence: 99%

The multiscale hybrid mixed method in general polygonal meshes

et al. 2020

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“…Among them are streamline upwind/Petrov-Galerkin method (SUPG) or Galerkin least squares method (GLS) [10,6], hp finite element methods [17,18], discontinuous Petrov-Galerkin methods (DPG) [8], residual-free bubble approaches (RFB) [2,5,4], methods with an additional non-linear diffusion [1], methods with stabilization by local orthogonal sub-scales [7] and hybridizable discontinuous Galerkin (HDG) methods [22]. Among the multiscale methods are variational multiscale methods (VMS) [13,15], multiscale finite element methods (MsFEM) [19,3], multiscale hybrid-mixed methods (MHM) [12] and local orthogonal decomposition methods (LOD) [9]. Specifically, the residual-based stabilization methods (SUPG, GLS and RFB) incorporate global stability properties into high accuracy in local regions away from boundary layers.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in two dimensions

Peterseim

Schedensack

2017

IMA Journal of Numerical Analysis

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We formulate a stabilized quasi-optimal Petrov-Galerkin method for singularly perturbed convection-diffusion problems based on the variational multiscale method. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on precomputed fine-scale correctors. The exponential decay of these correctors and their localisation to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, is rigorously justified. Under moderate assumptions, this stabilization guarantees stability and quasi-optimal rate of convergence for arbitrary mesh Péclet numbers on fairly coarse meshes at the cost of additional inter-element communication.

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“…Illustration of different interpolation spaces on edges. The one constant (left) and the two constants (right) cases [21].…”

Section: Numerical Validationmentioning

confidence: 99%

“…Recently, a new family of multiscale finite element methods, named Multiscale Hybrid-Mixed (MHM) method, was presented in [20] and further analyzed in [4]. The framework has since been extended to the linear elasticity model in [19] and the reactive-advectivediffusive problem in [21]. The MHM method has a notably general formulation that recovers some well-established finite element methods, such as the ones proposed in [12,6], under appropriate hypotheses.…”

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confidence: 99%

On the robustness of multiscale hybrid-mixed methods

Paredes¹,

Valentin

Versieux

2016

Math. Comp.

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In this work we prove uniform convergence of the Multiscale Hybrid-Mixed (MHM for short) finite element method for second order elliptic problems with rough periodic coefficients. The MHM method is shown to avoid resonance errors without adopting oversampling techniques. In particular, we establish that the discretization error for the primal variable in the broken H 1 and L 2 norms are O(h + ε δ ) and O(h 2 + h ε δ ), respectively, and for the dual variable is O(h + ε δ ) in the H(div; ·) norm, where 0 < δ ≤ 1/2 (depending on regularity). Such results rely on sharpened asymptotic expansion error estimates for the elliptic models with prescribed Dirichlet, Neumann or mixed boundary conditions. Flows in porous media, which commonly exhibit multiple scale structures, are usually modeled by a second order elliptic problem (Darcy equation) with rough discontinuous coefficients. Such a model arises when we consider the simulation of oil reservoirs in a highly heterogenous and/or fractured media. Multiscale problems necessarily require the use of very fine meshes, which makes their numerical approximation extremely expensive. Since the pioneering work of Babuska and Osborn [9] and its extension to higher dimensions by Hou and Wu [23], multiscale numerical methods have emerged as an attractive "divide and conquer" option to handle heterogeneous problems (see [15,14,36], just to cite a few). Overall, the idea relies on basis functions specially designed to upscale submesh oscillations to an overlying coarse mesh. As a result, such numerical method becomes precise on coarse meshes. Also interesting, the multiscale basis functions can be locally computed through completely independent problems. This makes the resulting numerical algorithm particularly attractive for use in parallel computing environments.

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On a Multiscale Hybrid-Mixed Method for Advective-Reactive Dominated Problems with Heterogeneous Coefficients

Cited by 31 publications

References 20 publications

The multiscale hybrid mixed method in general polygonal meshes

The multiscale hybrid mixed method in general polygonal meshes

Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in two dimensions

On the robustness of multiscale hybrid-mixed methods

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