Nonconforming Finite Element Scheme Research Articles

A Sequentially- Hybridized Locally Conservative Non-conforming Finite Element Scheme for Two-phase Flow Simulation through Heterogeneous Porous Media

Accurate two-phase flow modeling has a crucial role in detecting pollution in aquifers and/or in the simulating immiscible flow through porous media. In this research, a new hybrid numerical method based on the lower order non-conforming finite element method (NCFEM) and interior penalty discontinuous Galerkin (DG) method is proposed to simulate two-phase incompressible flow in heterogeneous porous media. The pressure and transport saturation equations are discretized by using the non-conforming Crouzeix-Raviart (CR) finite element and the symmetric weighed interior penalty discontinuous Galerkin (SWIPM). As a result of determining the degrees of freedom at the midpoint of neighboring element edges, a consistent set of pressure and velocity fields is obtained via the NCFEM. An H(div) projection based on the Raviart-Thomas (RT) element is implemented to improve the results resolution and preserve the continuity of the normal component of the velocity field. Besides, the spurious numerical oscillations of the saturation solution at the end of each time step are removed by using a novel vertex-based slope limiter, named the modified Chavent-Jaffre limiter. The accuracy and performance of the proposed numerical method are assessed by solving the Buckley-Leverett and McWhorter-Sunada benchmark problems, along with two test problems associated with highly heterogeneous porous media. The numerical results indicate that the proposed scheme has a remarkable potential to accurately capture the shock front and sharp interface zone of immiscible phases in the heterogeneous aquifers and other porous media.

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

This paper presents a nonconforming finite element scheme for the planar biharmonic equation, which applies piecewise cubic polynomials (P3) and possesses \({\cal O}({h^2})\) convergence rate for smooth solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H4 functions with \({\cal O}({h^2})\) accuracy in the broken H2 norm. Besides, a discrete strengthened Miranda-Talenti estimate (∇ 2h ·, ∇ 2h ·) = (∆h ·, ∆h ·), which is usually not true for nonconforming finite element spaces, is proved. The finite element space does not correspond to a finite element defined with Ciarlet’s triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.

A High Accuracy Nonconforming Finite Element Scheme for Helmholtz Transmission Eigenvalue Problem

In this paper, we consider a cubic $H^2$ nonconforming finite element scheme $B_{h0}^3$ which does not correspond to a locally defined finite element with Ciarlet$'$s triple but admit a set of local basis functions. For the first time, we deduce and write out the expression of basis functions explicitly. Distinguished from the most nonconforming finite element methods, $(\delta\Delta_h\cdot,\Delta_h\cdot)$ with non-constant coefficient $\delta>0$ is coercive on the nonconforming $B_{h0}^3$ space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the $B_{h0}^3$ scheme can provide $\mathcal{O}(h^2)$ approximation for the eigenspace in energy norm and $\mathcal{O}(h^4)$ approximation for the eigenvalues. We test the $B_{h0}^3$ scheme on the vary-coefficient bi-Laplace source and eigenvalue problem, further, transmission eigenvalue problem. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed scheme.

Analysis of a finite volume–finite element method for Darcy–Brinkman two-phase flows in porous media

In this paper, we are interested in the displacement of two incompressible phases in a Darcy–Brinkman flow in a porous media. The equations are obtained by the conservation of the mass and by considering the Brinkman regularization velocity of each phase. This model is treated in its general form with the whole nonlinear terms. This paper deals with construction and convergence analysis of a combined finite volume–nonconforming finite element scheme together with a phase-by-phase upstream according to the total velocity.

Numerical study of compositional compressible degenerate two-phase flow in saturated–unsaturated heterogeneous porous media

We study the convergence of a combined finite volume–nonconforming finite element scheme on general meshes for a partially miscible two-phase flow model in anisotropic porous media. This model includes capillary effects and exchange between the phases. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh. The relative permeability of each phase is decentered according to the sign of the velocity at the dual interface. The convergence of the scheme is proved thanks to an estimate on the two pressures which allows to show estimates on the discrete time and compactness results in the case of degenerate relative permeabilities. A key point in the scheme is to use particular averaging formula for the dissolution function arising in the diffusion term. We show also a simulation of hydrogen production in nuclear waste management. Numerical results are obtained by in-house numerical code.

Error analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems

The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite element discretization which converges owing to some a priori$$L^2$$L2 error estimates even for reduced regularity on non-convex polygonal domains. An equivalence result of that nonconforming finite element scheme to the mixed finite element method (MFEM) leads to the well-posedness of the discrete solution and to a priori error estimates for the MFEM. The explicit residual-based a posteriori error analysis allows some reliable and efficient error control and motivates some adaptive discretization which improves the empirical convergence rates in three computational benchmarks.

A combined finite volume–nonconforming finite element scheme for compressible two phase flow in porous media

We propose and analyze a combined finite volume---nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on its own pressure.

A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variableuand the auxiliary variableσwith respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

Superconvergence of a New Nonconforming Mixed Finite Element Scheme for Elliptic Problem

A new nonconforming mixed finite element scheme for the second-order elliptic problem is proposed based on a new mixed variational form. It has the lowest degrees of freedom on rectangular meshes. The superclose property is proven by employing integral identity technique. Then global superconvergence result is derived through interpolation postprocessing operators. At last, some numerical experiments are carried out to verify the theoretical analysis.

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

An anisotropic nonconforming finite element scheme with moving grids for parabolic integro-differential equations

A Crank-Nicolson scheme based on nonconforming finite element with moving grids is discussed for a class of parabolic integro-differential equations under anisotropic meshes. The corresponding convergence analysis is presented and the error estimates are obtained by using the interpolation operator instead of the conventional elliptic projection which is an indispensable tool in the convergence analysis of traditional finite element methods in previous literature.

A second order nonconforming rectangular finite element method for approximating Maxwell’s equations

The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell’s equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h3), properly one order higher than that of its interpolation error O(h2) in the broken energy norm, where h is the subdivision parameter tending to zero.

Two-Grid Finite Element Discretization Schemes Based on Shifted-Inverse Power Method for Elliptic Eigenvalue Problems

This paper discusses highly efficient discretization schemes for solving self-adjoint elliptic differential operator eigenvalue problems. Several new two-grid discretization schemes, including the conforming and nonconforming finite element schemes, are proposed by combining the finite element method with the shifted-inverse power method for matrix eigenvalue problems. With these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$ and the solution of a linear algebraic system on the fine grid $\pi_{h}$. Theoretical analysis shows that the schemes have a high efficiency. For instance, the resulting solution can maintain an asymptotically optimal accuracy by using the conforming linear element or the nonconforming Crouzeix–Raviart element by taking $H=O(\sqrt[4]{h})$. Numerical experiments are presented to support the theoretical analysis. In addition, this paper establishes multigrid discretization schemes and proves their efficiency.

A UNIFIED APPROACH TO MIMETIC FINITE DIFFERENCE, HYBRID FINITE VOLUME AND MIXED FINITE VOLUME METHODS

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 methods (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. 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.

Application of nonconforming finite elements to solving advection—diffusion problems

The paper investigates some nonconforming finite elements and nonconforming finite element schemes for solving an advection—diffusion equation. This investigation is aimed at finding new schemes for solving parabolic equations. The study uses a finite element method, variational-difference schemes, and test calculations. Two types of schemes are examined: one is obtained with the help of the Bubnov—Galerkin method from a weak problem determination (nonmonotone scheme), and the other one is a monotone up-stream scheme obtained from an approximate weak problem determination with a special approximation of the skew-symmetric terms.

An adaptive finite volume box scheme for solving a class of nonlinear parabolic equations

In this work we are interested in a fully nonconforming (Crouzeix–Raviart) finite element scheme for addressing a nonlinear parabolic equation. In addition to showing the equivalence with a finite volume box scheme we derive a posteriori error indicators in time and space, uncoupled as much as possible.

Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation

The equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media. The estimator is shown to provide a computable upper bound on the error and, up to a constant depending only on the geometry, provides two-sided bounds on the error. The robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly.

Advection schemes for unstructured grid ocean modelling

We study advection schemes for unstructured grid ocean models. Four linear advection schemes are investigated by solving a scalar transport equation. Schemes under consideration include continuous, non-conforming and discontinuous finite elements and finite volumes. A comprehensive derivation of the numerical schemes is presented and conservation and dispersion properties are discussed. An assessment is made by performing the test problem introduced by Hecht et al. [J. Geophys. Res. 100 (1995) 20763] in which a passive scalar field is advected through an analytical Stommel gyre. It is found that continuous finite elements and finite volumes have some difficulties to represent accurately solutions with steep gradients. As a result they are prone to generate unphysical oscillations. On the other hand, discontinuous and non-conforming finite element schemes perform better. This is due to their higher flexibility that makes them better suited to highly sheared flows.

Simulation of an Austenite-Twinned-Martensite Interface.

Developing numerical methods for predicting microstructure in materials is a large and important research area. Two examples of material microstructures are Austenite and Martensite. Austenite is a microscopic phase with simple crystallographic structure while Martensite is one with a more complex structure. One important task in materials science is the development of numerical procedures which accurately predict microstructures in Martensite. In this paper we present a method for simulating material microstructure close to an Austenite-Martensite interface. The method combines a quasi-Newton optimization algorithm and a nonconforming finite element scheme that successfully minimizes an approximation to the total stored energy near the interface of interest. Preliminary results suggest that the minimizers of this energy functional located by the developed numerical algorithm appear to display the desired characteristics.

Multigrid methods for a parameter dependent problem in primal variables

In this paper we consider multigrid methods for the parameter dependent problem of nearly incompressible materials. We construct and analyze multilevel-projection algorithms, which can be applied to the mixed as well as to the equivalent, non-conforming finite element scheme in primal variables. For proper norms, we prove that the smoothing property and the approximation property hold with constants that are independent of the small parameter. Thus we obtain robust and optimal convergence rates for the W-cycle and the variable V-cycle multigrid methods. The numerical results pretty well conform the robustness and optimality of the multigrid methods proposed.