Multiscale Hybrid-Mixed Method Research Articles

Revisiting the robustness of the multiscale hybrid-mixed method: The face-based strategy

This work proposes a new finite element for the multiscale hybrid-mixed method (MHM) applied to the Poisson equation with highly oscillatory coefficients. Unlike the original MHM method, multiscale bases are the solution to local Neumann problems driven by piecewise continuous polynomial interpolation on the skeleton faces of the macroscale mesh. As a result, we prove the optimal convergence of MHM by refining the face partition and leaving the mesh of macroelements fixed. This property allows the MHM method to be resonance free under the usual assumptions of local regularity. The numerical analysis of the method also revisits and complements the original approach proposed by D. Paredes, F. Valentin and H. Versieux (2017). Numerical experiments assess the new theoretical results.

A Petrov–Galerkin multiscale hybrid-mixed method for the Darcy equation on polytopes

A Petrov–Galerkin multiscale hybrid-mixed method for the Darcy equation on polytopes

A posteriori error estimator for a multiscale hybrid mixed method for Darcy's flows

AbstractThis article presents a computable and efficient procedure for a posteriori error estimations of approximate solutions of Darcy's flows given by a Multiscale Hybrid Mixed method. Based on a partition of the domain by polytopal macro subregions, this is a strategy for efficient simulations of problems with strongly varying solutions. The flux approximations interacting with the skeleton (subregion boundaries) are strongly constrained by a given trace space. Whilst the dimension of the piecewise polynomial trace space is expected to be as low as possible, the small‐scale features of the solution are supposed to be accurately resolved by completely independent local stable mixed solvers, the trace variable playing the role of Neumann boundary data. As the method already gives an equilibrated global H(div)‐conforming flux approximation, the methodology for the error estimation only requires a potential reconstruction. In addition to usual residual errors and indicators associated with the potential reconstruction, the estimation also takes into account the effect of discretizations of practical nonhomogeneous Dirichlet and Neumann boundary conditions. Based on the proposed a posteriori error estimator, ‐adaptive algorithms are constructed to guide a proper choice of the trace space. The performance of the error estimator and the adaptive scheme is numerically investigated through a set of illustrating test problems.

The MHM Method for Linear Elasticity on Polytopal Meshes

Abstract The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.

An adaptive multiscale hybrid-mixed method for the Oseen equations

A novel residual a posteriori error estimator for the Oseen equations achieves efficiency and reliability by including multilevel contributions in its construction. Originates from the Multiscale Hybrid Mixed (MHM) method, the estimator combines residuals from the skeleton of the first-level partition of the domain, along with the contributions from element-wise approximations. The second-level estimator is local and infers the accuracy of multiscale basis computations as part of the MHM framework. Also, the face-degrees of freedom of the MHM method shape the estimator and induce a new face-adaptive procedure on the mesh’s skeleton only. As a result, the approach avoids re-meshing the first-level partition, which makes the adaptive process affordable and straightforward on complex geometries. Several numerical tests assess theoretical results.

The multiscale hybrid mixed method in general polygonal meshes

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 $$k+1$$, $$k \ge 0$$, 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.

A Multiscale Hybrid-Mixed Method for the Helmholtz Equation in Heterogeneous Domains

This work proposes a novel multiscale finite element method for acoustic wave propagation in highly heterogeneous media which is accurate on coarse meshes. It originates from the primal hybridizati...

On a multiscalea posteriorierror estimator for the stokes and Brinkman equations

AbstractThis work proposes and analyzes a residual a posteriori error estimator for the multiscale hybrid-mixed (MHM) method for the Stokes and Brinkman equations. The error estimator relies on the multi-level structure of the MHM method and considers two levels of approximation of the method. As a result the error estimator accounts for a first-level global estimator defined on the skeleton of the partition and second-level contributions from element-wise approximations. The analysis establishes local efficiency and reliability of the complete multiscale estimator. Also, it yields a new face-adaptive strategy on the mesh’s skeleton, which avoids changing the topology of the global mesh. Specially designed to work on multiscale problems, the present estimator can leverage parallel computers since local error estimators are independent of each other. Academic and realistic multiscale numerical tests assess the performance of the estimator and validate the adaptive algorithms.

A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers

Multiscale Hybrid Mixed (MHM) method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions. For fluid flow, normal fluxes (multiplier) over macro element boundaries, and coarse piecewise constant potential approximations in each macro element are computed (upscaling). Then, small details are resolved by local problems, using fine representations inside the macro elements, setting the multiplier as Neumann boundary conditions (downscaling). In this work a variant of the method is developed, denoted by MHM- H(div), adopting mixed finite elements at the downscaling stage, instead of continuous finite elements used in all previous publications of the method. Thus, this alternative MHM method inherits improvements typical of mixed methods, as better flux accuracy, and local mass conservation at the micro scale level inside the macro elements which are important properties for multi-phase flows in rough heterogeneous media. Different two-scale stable space settings are considered. Vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries. In each macro element, the internal flux components, with vanishing normal traces, and the potential approximations, may be enriched in different extents: with respect to internal mesh size, internal polynomial degree, or both, the choice being determined by the problem at hands. A unified general error analysis of the MHM-H(div) method is presented for all these two-scale space scenarios. Both MHM versions are compared for 2D test problems, with smooth solutions, for convergence rates verification, and for Darcy’s flow in heterogeneous media. MHM-H(div) 3D simulations are presented for a known singular Darcy’s solution, using adaptive macro partitions, and for an oscillatory permeability scenario.

The multiscale hybrid-mixed method for the maxwell equations in heterogeneous media

The multiscale hybrid-mixed method for the maxwell equations in heterogeneous media

The Multiscale Hybrid-Mixed method for the Maxwell Equations in Heterogeneous Media

In this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for the two-and three-dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse-mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a divide-and-conquer framework. Several numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy to solving heterogeneous media problems on coarse meshes.

Multiscale hybrid-mixed method for the Stokes and Brinkman equations—The method

The multiscale hybrid-mixed (MHM) method is extended to the Stokes and Brinkman equations with highly heterogeneous coefficients. The approach is constructive. We first propose an equivalent dual-hybrid formulation of the original problem using a coarse partition of the heterogeneous domain. Faces may be not aligned with jumps in the data. Then, the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent Stokes (or Brinkman) problems at the continuous level. Owing to this decomposition, the one-level MHM method stems from the standard Galerkin approach for the Lagrange multiplier space. Basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation. They are the exact solution of the local problems with prescribed Neumann boundary conditions on faces driven by the Lagrange multipliers. We make the MHM method effective by adopting the unusual stabilized finite element method to solve the local problems approximately. As such, equal-order interpolation turns out to be an option for the velocity, the pressure and the Lagrange multipliers. The numerical solutions share the important properties of the continuum, such as local equilibrium with respect to external forces and local mass conservation. Several academic and highly heterogeneous tests infer that the method achieves super-convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms.

FV-MHMM method for reservoir modeling

The present paper proposes a new family of multiscale finite volume methods. These methods usually deal with a dual mesh resolution, where the pressure field is solved on a coarse mesh, while the saturation fields, which may have discontinuities, are solved on a finer reservoir grid, on which petrophysical heterogeneities are defined. Unfortunately, the efficiency of dual mesh methods is strongly related to the definition of up-gridding and down-gridding steps, allowing defining accurately pressure and saturation fields on both fine and coarse meshes and the ability of the approach to be parallelized. In the new dual mesh formulation we developed, the pressure is solved on a coarse grid using a new hybrid formulation of the parabolic problem. This type of multiscale method for pressure equation called multiscale hybrid-mixed method (MHMM) has been recently proposed for finite elements and mixed-finite element approach (Harder et al. 2013). We extend here the MH-mixed method to a finite volume discretization, in order to deal with large multiphase reservoir models. The pressure solution is obtained by solving a hybrid form of the pressure problem on the coarse mesh, for which unknowns are fluxes defined on the coarse mesh faces. Basis flux functions are defined through the resolution of a local finite volume problem, which accounts for local heterogeneity, whereas pressure continuity between cells is weakly imposed through flux basis functions, regarded as Lagrange multipliers. Such an approach is conservative both on the coarse and local scales and can be easily parallelized, which is an advantage compared to other existing finite volume multiscale approaches. It has also a high flexibility to refine the coarse discretization just by refinement of the lagrange multiplier space defined on the coarse faces without changing nor the coarse nor the fine meshes. This refinement can also be done adaptively w.r.t. a posteriori error estimators. The method is applied to single phase (well-testing) and multiphase flow in heterogeneous porous media

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

A new family of finite element methods, named Multiscale Hybrid-Mixed method (or MHM for short), aims to solve reactive-advective dominated problems with multiscale coefficients on coarse meshes. The underlying upscaling procedure transfers to the basis functions the responsibility of achieving high orders of accuracy. The upscaling is built inside the general framework of hybridization, in which the continuity of the solution is relaxed a priori and imposed weakly through the action of Lagrange multipliers. This characterizes the unknowns as the solutions of local problems with Robin boundary conditions driven by the multipliers. Such local problems are independent of one another, yielding a process naturally shaped for parallelization and adaptivity. Moreover, the multiscale decomposition indicates a new adaptive algorithm to set up local spaces defined using a face-based a posteriori error estimator. Interestingly, it also embeds a postprocessing of the dual variable (flux) which preserves local conservation properties of the exact solution. Extensive numerical validations assess the claimed optimal rates of convergence, the robustness of the method with respect to the model's coefficients, and the adaptivity algorithm.

Abstract multiscale-hybrid-mixed methods

In an abstract setting, we investigate the basic ideas behind the Multiscale Hybrid Mixed (MHM) method, a Domain Decomposition scheme designed to solve multiscale partial differential equations (PDEs) in parallel. As originally proposed, the MHM method starting point is a primal hybrid formulation, which is then manipulated to result in an efficient method that is based on local independent PDEs and a global problem that is posed on the skeleton of the finite element mesh. Recasting the MHM method in a more general framework, we investigate some conditions that yield a well-posed method. We apply the general ideas to different formulations, and, in particular, come up with an interesting and fruitful connection between the Multiscale Finite Element Method and a dual hybrid method. Finally, we propose a method that combines the main ideas of the Discontinuous Enrichment Method and the MHM method.

Multiscale Hybrid-Mixed Method

This work presents a priori and a posteriori error analyses of a new multiscale hybrid-mixed method (MHM) for an elliptic model. Specially designed to incorporate multiple scales into the construction of basis functions, this finite element method relaxes the continuity of the primal variable through the action of Lagrange multipliers, while assuring the strong continuity of the normal component of the flux (dual variable). As a result, the dual variable, which stems from a simple postprocessing of the primal variable, preserves local conservation. We prove existence and uniqueness of a solution for the MHM method as well as optimal convergence estimates of any order in the natural norms. Also, we propose a face-residual a posteriori error estimator, and prove that it controls the error of both variables in the natural norms. Several numerical tests assess the theoretical results.