We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.
The gradient discretisation method for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differences
Monash University（莫纳什大学）；Indian Institute of Technology Bombay（印度理工学院孟买分校）
We present a hybrid mimetic finite-difference and virtual element formulation for coupled single-phase poromechanics on unstructured meshes. The key advantage of the scheme is that it is convergent on complex meshes containing highly distorted cells with arbitrary shapes. We use a local pressure-jump stabilization method based on unstructured macro-elements to prevent the development of spurious pressure modes in incompressible problems approaching undrained conditions. A scalable linear solution strategy is obtained using a block-triangular preconditioner designed specifically for the saddle-…
Upwinded mass fluxes are described and analysed for advection operators discretised using mixed mimetic spectral elements. This involves a Petrov-Galerkin formulation by which the mass flux test functions are evaluated at downstream locations along velocity characteristics. As for the original mixed mimetic spectral element advection operator, the upwinded mass flux advection operator is conservative, however unlike the original advection operator, which is purely hyperbolic, the upwinded advection operator adds dissipation which is biased towards high wave numbers. The upwinded advection oper…
A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are disconti…
A decoupled energy-stable mixed finite element method for Poisson-Nernst-Planck-Navier-Stokes equations
School of Mathematics and Computational Science, Guilin University of Electronic Technology（桂林电子科技大学数学与 computational 科学学院）；Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation（广西高校数据分析与计算重点实验室）；Guangxi Applied Mathematics Center (GUET)（广西应用数学中心（桂林电子科技大学））
We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.
A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach global systems with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the s…
The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non In…
We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively …
The hybrid Raviart-Thomas mixed discontinuous Galerkin (HRTMDG) method is proposed for solving the Helmholtz equation. With a new energy norm, we establish the existence and uniqueness of the HRTMDG method, and give its convergence analysis. The corresponding error estimate shows that the HRTMDG method has an optimal $L^2$-norm convergence accuracy which is independent of wavenumber.
For the planar Navier--Lamé equation in mixed form with symmetric stress tensors, we prove the uniform quasi-optimal convergence of an adaptive method based on the hybridized mixed finite element proposed in [Gong, Wu, and Xu: Numer.~Math., 141 (2019), pp.~569--604]. The main ingredients in the analysis consist of a discrete a posteriori upper bound and a quasi-orthogonality result for the stress field under the mixed boundary condition. Compared with existing adaptive methods, the proposed adaptive algorithm could be directly applied to the traction boundary condition and be easily implemente…
We introduce and analyze a stress-based formulation for Zener's model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition. We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness of the problem through a standard Galerkin procedure. We propose and analyze mixed continuous and discontinuous Galerkin space discretizations of the problem and derive optimal error bounds for ea…
Wormhole propagation plays a very important role in the product enhancement of oil and gas reservoir. A new combined hybrid mixed finite element method is proposed to solve incompressible wormhole propagation problem with discontinuous Galerkin finite element procedure, in which, the new hybrid mixed finite element algorithm is established for pressure equation, while the discontinuous Galerkin finite element method is considered for concentration equation, and then the porosity function is computed straightly by the approximate value of the concentration. This new combined method can keep loc…
A new hybrid algorithm for LDU-factorization for large sparse matrix combining iterative solver, which can keep the same accuracy as the classical factorization, is proposed. The last Schur complement will be generated by iterative solver for multiple right-hand sides using block GCR method with the factorization in lower precision as a preconditioner, which achieves mixed precision arithmetic, and then the Schur complement will be factorized in higher precision. In this algorithm, essential procedure is decomposition of the matrix into a union of moderate and hard parts, which is realized by …
In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We intro…
Analysis of hybrid methods of mixed-shear-projected triangular and quadrilateral elements for Reissner-Mindlin plates
School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）；School of Mathematics, Sichuan University（四川大学数学学院）；Experiment Center, China West Normal University（西华师范大学实验中心）
This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdom…
In this work we propose a novel block preconditioner, labelled Explicit Decoupling Factor Approximation (EDFA), to accelerate the convergence of Krylov subspace solvers used to address the sequence of non-symmetric systems of linear equations originating from flow simulations in porous media. The flow model is discretized blending the Mixed Hybrid Finite Element (MHFE) method for Darcy's equation with the Finite Volume (FV) scheme for the mass conservation. The EDFA preconditioner is characterized by two features: the exploitation of the system matrix decoupling factors to recast the Schur com…
We consider the quasi-static Biot's consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement $\boldsymbol{u}$ of the solid matrix, the seepage velocity $\boldsymbol{v}$ of the fluid and the pore pressure $p$. As conservation of fluid mass is a leading physical principle in poromechanics, we preserve this property using an $\boldsymbol{H}(\operatorname{div})$-conforming ansatz for $\boldsymbol{u}$ and $\boldsymbol{v}$ together with an appropriate pressure space. This results in Stokes and Darcy stability and exact, that is, p…
A Hybrid High-Order method for the Cahn-Hilliard problem in mixed form
University of Montpellier, Institut Montpelliérain Alexander Grothendieck（蒙彼利埃大学，亚历山大·格罗滕迪克蒙彼利埃研究所）；INRIA Lemon team（法国国家信息与自动化研究所柠檬团队）；Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain（玻璃与界面表面，CNRS/圣戈班联合研究实验室125）
When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse pr…
BDDC for Mixed-Hybrid Formulation of Flow in Porous Media with Combined Mesh Dimensions
Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Technical University of Liberec（利贝雷茨理工大学）；University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）
We present a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in a fractured porous media. In this model, the fracture is treated as an $(d-1)$-dimensional interface within the $d$-dimensional fractured porous domain, for $d=2, 3$. Two classes of fracture are distinguished based on the permeability magnitude ratio between the fracture and its surrounding medium: when the permeability in the fracture is (significantly) larger than in its surrounding medium, it is considered as a {\it conductive} fracture; when the permeability in the fracture is…
This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set …
This paper presents a novel numerical method for the hybrid reliability analysis by using the uncertainty theory. Aleatory uncertainty and epistemic uncertainty are considered simultaneously in this method. Epistemic uncertainty is characterized by the uncertainty theory, and the effect of epistemic uncertainty is quantified by the sub-additive uncertain measure. Then, under the framework of the chance theory which can be interpreted as the combination of the probability theory and the uncertainty theory, a general uncertainty quantification model is established to deal with the hybrid reliabi…
Due to the wide separation of time scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves, while not suffering from severe restrictions on the timestep size. To propagate the state of the atmosphere forward in time, a non-linear equation for the prognostic variables has to be solved at every timestep. Since the nonlinearity is typically weak, this is done with a small number of Newton- or Picard- iterations, which in turn require the efficient sol…
