Mixed and nonconforming finite element methods on a system of polygons

Vohralı́k, Martin; Maryška, Jiří; Severýn, Otto

doi:10.1016/j.apnum.2006.02.005

Cited by 36 publications

(39 citation statements)

References 12 publications

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“…When solving fluid flow in fracture networks, a notable difference between 2-D and 3-D models is the geometric meshing (Vohralik et al 2007;Benedetto et al 2016a). In a 2-D model, the fractures are commonly treated as line segments for which the meshing is considerably simple; however, in a 3-D model, especially when the local variation of aperture in fractures is considered for solving the local cubic law, the fractures are represented by planes that have zero to several intersections with others, which complicates the structure of meshing (Benedetto et al 2016b).…”

Section: Three-dimensional Fracture Networkmentioning

confidence: 98%

Review: Mathematical expressions for estimating equivalent permeability of rock fracture networks

et al. 2016

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Fracture networks play a more significant role in conducting fluid flow and solute transport in fractured rock masses, comparing with that of the rock matrix. Accurate estimation of the permeability of fracture networks would help researchers and engineers better assess the performance of projects associated with fluid flow in fractured rock masses. This study provides a review of previous works that have focused on the estimation of equivalent permeability of twodimensional (2-D) discrete fracture networks (DFNs) considering the influences of geometric properties of fractured rock masses. Mathematical expressions for the effects of nine important parameters that significantly impact on the equivalent permeability of DFNs are summarized, including (1) fracturelength distribution, (2) aperture distribution, (3) fracture surface roughness, (4) fracture dead-end, (5) number of intersections, (6) hydraulic gradient, (7) boundary stress, (8) anisotropy, and (9) scale. Recent developments of 3-D fracture networks are briefly reviewed to underline the importance of utilizing 3-D models in future research.

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Section: Three-dimensional Fracture Networkmentioning

confidence: 98%

Review: Mathematical expressions for estimating equivalent permeability of rock fracture networks

et al. 2016

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“…We recall that the Crouzeix-Raviart nonconforming finite element method is equivalent to the lowest-order Raviart-Thomas mixed finite element method [10], see, e.g., Arnold and Brezzi [3], Marini [9], Arbogast and Chen [2], and [15,17], and the references therein.…”

Section: The Crouzeix-raviart Nonconforming Finite Element Methodsmentioning

confidence: 99%

From face to element unknowns by local static condensation with application to nonconforming finite elements

Vohralı́k

Wohlmuth

2013

Computer Methods in Applied Mechanics and Engineering

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We derive in this paper a new local static condensation strategy which allows to reduce significantly the number of unknowns in algebraic systems arising in discretization of partial differential equations. We apply it to the discretization of a model linear elliptic diffusion and a model nonlinear parabolic advection-diffusion-reaction problem by Crouzeix-Raviart nonconforming finite elements. Herein, the unknowns, originally associated with the mesh faces, can be reduced to new unknowns associated with the mesh elements. The resulting matrices are sparse, with possibly only four nonzero entries per row in two space dimensions, positive definite in dependence on the mesh geometry and the diffusion-dispersion tensor, but in general nonsymmetric. Our approach consists in introducing new element unknowns, the identification of suitable local vertex-based subproblems, and the inversion of the corresponding local matrices. We give sufficient conditions for the well-posedness of the local problems, as well as for the resulting global one. In addition, we provide a geometrical interpretation which suggests how to influence the form of the local and global matrices depending on the local mesh and data. We finally present an abstract generalization allowing for a further reduction of the number of unknowns, typically to one unknown per a set of mesh elements. We conclude by numerical experiments which show that the condition number of the resulting matrices is robust with respect to the mesh anisotropies and the diffusion tensor inhomogeneities.

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“…Some complementary results are presented by Marini and Pietra [45] and in [25] and [8]. Links between the mixed finite element and nonconforming finite element methods are then, in particular, given in [9,25,8], Marini [46], Chen [26], or [37,65]. Recently, Cockburn and Gopalakrishnan [28,29] showed that analysis of mixed methods can be entirely based on the hybridization (cf.…”

Section: Introductionmentioning

confidence: 99%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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Mixed and nonconforming finite element methods on a system of polygons

Cited by 36 publications

References 12 publications

Review: Mathematical expressions for estimating equivalent permeability of rock fracture networks

Review: Mathematical expressions for estimating equivalent permeability of rock fracture networks

From face to element unknowns by local static condensation with application to nonconforming finite elements

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

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