We investigate a mortar technique for mixed finite element approximations of Darcy flow on non-matching grids in which the normal flux is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of pressure. In the mixed formulation of the problem, the normal flux is an essential boundary condition and it is incorporated with the use of suitable extension operators. Two such extension operators are considered and we analyze the resulting formulations with respect to stability and convergence. We further generalize the theoretical results, showing that the same domain decomposition technique is applicable to a class of saddle point problems satisfying mild assumptions. An example of coupled Stokes-Darcy flows is presented.
The purpose of this work is to study mortar methods for linear elasticity using standard low order finite element spaces. Based on residual stabilization, we introduce a stabilized mortar method for linear elasticity and compare it to the unstabilized mixed mortar method. For simplicity, both methods use a Lagrange multiplier defined on a trace mesh inherited from one side of the interface only. We derive a quasi-optimality estimate for the stabilized method and present the stability criteria of the mixed $P_1-P_1$ approximation. Our numerical results demonstrate the stability and the converge…
The flux-mortar mixed finite element method was recently developed for a general class of domain decomposition saddle point problems on non-matching grids. In this work we develop the method for Darcy flow using the multipoint flux approximation as the subdomain discretization. The subdomain problems involve solving positive definite cell-centered pressure systems. The normal flux on the subdomain interfaces is the mortar coupling variable, which plays the role of a Lagrange multiplier to impose weakly continuity of pressure. We present well-posedness and error analysis based on reformulating …
We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with non-matching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability, as well as a priori error estimates for the spat…
The paper concerns the modelling of masonry elements strengthened through Textile Reinforced Mortar (TRM), a near surface system made of fiber-based grids or textiles embedded in mortar layers. Recently, the author, focusing on the mechanical characterization of TRM composites, developed a detailed level modelling approach by using the free, open-source Finite-Element code OOFEM, for the simulation of experimental tests on TRM coupons (pull-out tests, tensile tests, shear bond tests and in-plane shear tests). The model was capable to account for the failure of single components (e.g. the fiber…
AlgMortar: a fully algebraic multiscale mortar preconditioner
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算科学学院）；Piri Technologies, LLC（Piri科技有限公司）；Department of Mathematical Sciences, The University of Texas at Dallas（德克萨斯大学达拉斯分校数学科学系）
A method to treat frictional contact problems along embedded surfaces in the finite element framework is developed. Arbitrarily shaped embedded surfaces, cutting through finite element meshes, are handled by the X-FEM. The frictional contact problem is solved using the monolithic augmented Lagrangian method within the mortar framework which was adapted for handling embedded surfaces. We report that the resulting mixed formulation is prone to mesh locking in case of high elastic and mesh density contrasts across the contact interface. The mesh locking manifests itself in spurious stress oscilla…
This work proposes a novel model and numerical formulation for lubricated contact problems describing the mutual interaction between two deformable 3D solid bodies and an interposed fluid film. The solid bodies are consistently described based on nonlinear continuum mechanics allowing for finite deformations and arbitrary constitutive laws. The fluid film is modelled as a quasi-2D flow problem on the interface between the solids governed by the averaged Reynolds equation. The averaged Reynolds equation accounts for surface roughness utilizing spatially homogenized, effective fluid parameters a…
This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.
A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H({\rm div})-P_k (k\geq 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, which can be viewed as the Hu-Zhang elements enriched with higher order interior bubble functions. The displacement field is approximated by $C^{-1}-P_{k-1}$ polynomial vectors enriched with higher order terms to ensure the stability condition. For both the proposed mixed elements and their mass lumping schemes, optimal …
For the second lowest-order Raviart--Thomas mixed method, we prove that the canonical interpolant and finite element solution for the vector variable in elliptic problems are superclose in the $H(\text{div})$-norm on mildly structured meshes, where most pairs of adjacent triangles form approximate parallelograms. We then develop a family of postprocessing operators for Raviart--Thomas mixed elements on triangular grids by using the idea of local least squares fittings. Super-approximation property of the postprocessing operators for the lowest and second lowest order Raviart--Thomas elements i…
A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids
We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element method for Darcy flow, the method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the vertex quadrature rule in order to localize the interaction of degrees of freedom. This allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell-centered system for displa…
This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress $\bmσ=-\nabla^{2}u$ is sought in the Sobolev space $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ simultaneously with the displacement $u$ in $L^{2}(Ω)$. Stemming from the structure of $H(\bm{div},Ω;\mathbb{S})$ conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ conforming finite element spaces are constructed by imposing the normal continuity of $\bm{div}\bmσ$…
The numerical simulation of physical processes in the underground frequently entails challenges related to the geometry and/or data. The former are mainly due to the shape of sedimentary layers and the presence of fractures and faults, while the latter are connected to the properties of the rock matrix which might vary abruptly in space. The development of approximation schemes has recently focused on the overcoming of such difficulties with the objective of obtaining numerical schemes with good approximation properties. In this work we carry out a numerical study on the performances of the Mi…
We develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method is developed on simplicial grids in [4]. The method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation…
Nonlocal multicontinuum (NLMC) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models
Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；The Chinese University of Hong Kong (CUHK)（香港中文大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
In many applications the accurate representation of the computational domain is a key factor to obtain reliable and effective numerical solutions. Curved interfaces, which might be internal, related to physical data, or portions of the physical boundary, are often met in real applications. However, they are often approximated leading to a geometrical error that might become dominant and deteriorate the quality of the results. Underground problems often involve the motion of fluids where the fundamental governing equation is the Darcy law. High quality velocity fields are of paramount importanc…
In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The co…
Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution ha…
Stabilized mixed finite element methods for linear elasticity on simplicial grids in $\mathbb{R}^{n}$
University of California at Irvine（加州大学尔湾分校）；LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院和线性代数及其应用研究中心）；College of Mathematics and Information Science, Wenzhou University（温州大学数学与信息工程学院）
The capability to incorporate moving geometric features within models for complex simulations is a common requirement in many fields. Fluid mechanics within aeronautical applications, for example, routinely feature rotating (e.g. turbines, wheels and fan blades) or sliding components (e.g. in compressor or turbine cascade simulations). With an increasing trend towards the high-fidelity modelling of these cases, in particular combined with the use of high-order discontinuous Galerkin methods, there is therefore a requirement to understand how different numerical treatments of the interfaces bet…
