Raviart-Thomas Elements Research Articles

Superconvergence analysis of an [formula omitted]-Galerkin mixed finite element method for Sobolev equations

An H1-Galerkin mixed finite element method (MFEM) is discussed for the Sobolev equations with the bilinear element and zero order Raviart–Thomas element (Q11+Q10×Q01). The existence and uniqueness of the solutions about the approximation scheme are proved. Two new important lemmas are given by using the properties of the integral identity and the Bramble–Hilbert lemma, which lead to the superclose results of order O(h2) for original variable u in H1 norm and flux q→ in H(div;Ω) norm under semi-discrete scheme. Furthermore, two new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained. On the other hand, a second order fully-discrete scheme with superclose property O(h2+τ2) is also proposed. At last, numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.

Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text], and by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text]. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise [Formula: see text] polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element [Formula: see text] plus [Formula: see text] in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

Analysis of an augmented mixed-FEM for the Navier-Stokes problem

In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a “nonlinear-pseudostress” tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinearpseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equation and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach's fixed point Theorem and Lax-Milgram's Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish wellposedness and the corresponding Cea's estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k+1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided. This contribution is based on joint work with Ricardo Oyarzua (Universidad del Bio-Bio) and Giordano Tierra (Charles University).

Convergence of an adaptive lowest-order Raviart–Thomas element method for general second-order linear elliptic problems

Convergence of an adaptive lowest-order Raviart–Thomas element method for general second-order linear elliptic problems

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier---Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the standard mixed formulation in the Navier---Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.

A priori error analysis of the BEM with graded meshes for the electric field integral equation on polyhedral surfaces

The Galerkin boundary element discretisations of the electric field integral equation (EFIE) on Lipschitz polyhedral surfaces suffer slow convergence rates when the underlying surface meshes are quasi-uniform and shape-regular. This is due to singular behaviour of the solution to this problem in neighbourhoods of vertices and edges of the surface. Aiming to improve convergence rates of the Galerkin boundary element method (BEM) for the EFIE on a Lipschitz polyhedral closed surface Γ, we employ anisotropic meshes algebraically graded towards the edges of Γ. We prove that on sufficiently graded meshes the h-version of the BEM with the lowest-order Raviart–Thomas elements regains (up to a small order of ε>0) an optimal convergence rate (i.e., the rate of the h-BEM on quasi-uniform meshes for smooth solutions).

The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on [formula omitted]-rectangular grids

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress and the displacement. The discrete spaces for the normal stress τii, the shear stress τij and the displacement ui are span{1,xi}, span{1,xi,xj} and span{1}, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, the element is nothing else but the 1D Raviart–Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element are 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. These elements are the simplest element for any space dimension.The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved. Numerical tests in 2D and 3D are presented to show a superiority of the new elements over others, as a superconvergence is exhibited and proved.

Approximation of the time-dependent induction equation with advection using Whitney elements

Purpose – The purpose of this paper is to present a new formulation for taking into account the convective term due to an imposed velocity field in the induction equation in a code based on Whitney elements called DOLMEN. Different Whitney forms are used to approximate the dependent variables. The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. DOLMEN is developed to investigate the dynamo action in non-axisymmetric domains like the impeller driven flow of the von Kármán Sodium (VKS) experiment. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field. Design/methodology/approach – Different Whitney forms are used to approximate the dependent variables. The vector potential is discretized using first-order edge elements of the first family. The velocity is approximated by using the first-order Raviart-Thomas elements. The time stepping is done by using the Crank-Nicolson scheme. Findings – The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field. Originality/value – The findings offer a basis to a scenario for the VKS dynamo.

A primal-mixed formulation for the strong coupling of quasi-Newtonian fluids with porous media

In this work we analyze a primal-mixed finite element method for the coupling of quasi-Newtonian fluids with porous media in 2D and 3D. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions on the interface between the fluid and the porous medium are given by mass conservation, balance of normal forces and the Beavers-Joseph-Saffman law. We apply a primal formulation in the Stokes domain and a mixed formulation in the Darcy formulation. The "strong coupling" concept means that the conservation of mass across the interface is introduced as an essential condition in the space where the velocity unknowns live. In this way, under some assumptions on the nonlinear kinematic viscosity, a generalization of the Babuska-Brezzi theory is utilized to show the well posedness of the primal-mixed formulation. Then, we introduce a Galerkin scheme in which the discrete conservation of mass is imposed approximately through an orthogonal projector. The unique solvability of this discrete system and its Strang-type error estimate follow from the generalized Babuska-Brezzi theory as well. In particular, the feasible finite element subspaces include Bernadi-Raugel elements for the Stokes flow, and either the Raviart-Thomas elements of lowest order or the Brezzi-Douglas-Marini elements of first order for the Darcy flow, which yield nonconforming and conforming Galerkin schemes, respectively. In turn, piecewise constant functions are employed to approximate in both cases the global pressure field in the Stokes and Darcy domain. Finally, several numerical results illustrating the good performance of both discrete methods and confirming the theoretical rates of convergence, are provided.

Primal-mixed formulations for reaction–diffusion systems on deforming domains

We propose a finite element formulation for a coupled elasticity–reaction–diffusion system written in a fully Lagrangian form and governing the spatio-temporal interaction of species inside an elastic, or hyper-elastic body. A primal weak formulation is the baseline model for the reaction–diffusion system written in the deformed domain, and a finite element method with piecewise linear approximations is employed for its spatial discretization. On the other hand, the strain is introduced as mixed variable in the equations of elastodynamics, which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction–diffusion system written in a deformed domain. The discrete mechanical problem yields a mixed finite element scheme based on row-wise Raviart–Thomas elements for stresses, Brezzi–Douglas–Marini elements for displacements, and piecewise constant pressure approximations. The application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed, and some illustrative examples are provided and extensively analyzed.

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

SummaryThis paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity, and pressure, for the Brinkman problem with nonstandard boundary conditions. The approach is based on the introduction of Galerkin least‐squares terms arising from the constitutive equation relating the aforementioned unknowns and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic, which, thanks to the Lax–Milgram theorem, and besides proving the well‐posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart–Thomas elements of any order for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle‐bubble and edge‐bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators and showing the behavior of the associated adaptive algorithms, are reported. Copyright © 2015 John Wiley & Sons, Ltd.

Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements

We propose an effective and flexible way to assemble finite element stiffness and mass matrices in MATLAB. We apply this for problems discretized by edge finite elements. Typical edge finite elements are Raviart–Thomas elements used in discretizations of H(div) spaces and Nédélec elements in discretizations of H(curl) spaces. We explain vectorization ideas and comment on a freely available MATLAB code which is fast and scalable with respect to time.

A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations

In this work, we study the Brinkman-Forchheimer equations for unsteady flows. We prove the continuous dependence of the solution on the Brinkman's and Forchheimer's coefficients as well as the initial data and external forces. Next, we propose and study a perturbed compressible system that approximate the Brinkman-Forchheimer equations. Finally, we propose a time discretization of the perturbed system by a semi-implicit Euler scheme and next a lowest-order Raviart-Thomas element is applied for spatial discretization. Some numerical results are given.

Comparison of pressure and displacement formulations for finite elements in linear time-harmonic acoustics

In this article, we compare the finite element solution of the pressure and the displacement formulations for the acoustic problem in the frequency domain. The Helmholtz differential equation is solved for both, the spectral and the source problems, and each of these is formulated in terms of pressure and displacement. The pressure formulation allows to use standard Lagrangian elements whereas Raviart–Thomas elements are applied for the displacement formulation. Both formulations are tested in several examples. It is shown that Raviart–Thomas elements account for a reasonable alternative to Lagrangian elements in some cases.

Analysis of an augmented pseudostress-based mixed formulation for a nonlinear Brinkman model of porous media flow

Abstract In this paper we introduce and analyze an augmented mixed finite element method for the two-dimensional nonlinear Brinkman model of porous media flow with mixed boundary conditions. More precisely, we extend a previous approach for the respective linear model to the present nonlinear case, and employ a dual-mixed formulation in which the main unknowns are given by the gradient of the velocity and the pseudostress. In this way, and similarly as before, the original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, a feasible choice of finite element subspaces is given by Raviart–Thomas elements of order k ≥ 0 for the pseudostress, piecewise polynomials of degree ≤ k for the gradient of the velocity, and continuous piecewise polynomials of degree ≤ k + 1 for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behavior of the associated adaptive algorithm, are provided.

Verification of functional a posteriori error estimates for obstacle problem in 2D

We verify functional a posteriori error estimates proposed by S. Repin for a class of obstacle problems in two space dimensions. New benchmarks with known analytical solution are constructed based on one dimensional benchmark introduced by P. Harasim and J. Valdman. Numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. Error of the approximation is measured by a functional majorant. The majorant value contains three unknown fields: a gradient field discretized by Raviart–Thomas elements, Lagrange multipliers field discretized by piecewise constant functions and a scalar parameter β. The minimization of the majorant value is realized by an alternate minimization algorithm, whose convergence is discussed. Numerical results validate two estimates, the energy estimate bounding the error of approximation in the energy norm by the difference of energies of discrete and exact solutions and the majorant estimate bounding the difference of energies of discrete and exact solutions by the value of the functional majorant.

The Enriched Crouzeix–Raviart Elements are Equivalent to the Raviart–Thomas Elements

For both the Poisson model problem and the Stokes problem in any dimension, this paper proves that the enriched Crouzeix-Raviart elements are actually identical to the first order Raviart-Thomas elements in the sense that they produce the same discrete stresses. This result improves the previous result in literature which, for two dimensions, states that the piecewise constant projection of the stress by the first order Raviart-Thomas element is equal to that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace operator, this paper proves that the error of the enriched Crouzeix-Raviart element is equivalent to that of the Raviart-Thomas element up to higher order terms.

A finite element analysis of a pseudostress formulation for the Stokes eigenvalue problem

In this paper we analyse a finite element approximation of the Stokes eigenvalue problem. We introduce a variational formulation relying only on the pseudostress tensor and propose a discretization by means of the lowest-order Brezzi–Douglas–Marini mixed finite element. However, similar results hold true for other H(div)-conforming elements, like Raviart–Thomas elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates. Finally, we report some numerical tests supporting our theoretical results.

Existence and approximation of a mixed formulation for thin film magnetization problems in superconductivity

We recall a recently introduced mixed formulation of thin film magnetization problems for type-II superconductors written in terms of two variables, the electric field and the magnetization function, see [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. A finite element approximation, [Formula: see text], based on this mixed formulation, involving the lowest-order Raviart–Thomas element for approximating the electric field, was also introduced in [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. Here h, τ are the spatial and temporal discretization parameters, and [Formula: see text] with p-1 the value of power in the current–voltage relation characterizing the superconducting material. In this paper, we establish well-posedness of [Formula: see text], and prove convergence of the unique solution of [Formula: see text] to a solution of the power law model ( Q r), for a fixed r > 1, as h, τ → 0. In addition, we prove convergence of a solution of ( Q r) to a solution of the critical state model (Q), as r → 1. Hence, we prove existence of solutions to ( Q r), for a fixed r > 1, and (Q). Finally, numerical experiments are presented.

Equal order approximations enriched with bubbles for coupled Stokes–Darcy problem

As a remedy to the instability of the Galerkin finite element formulation, symmetric stabilization techniques such as the continuous interior penalty, the subgrid and local projection methods were proposed and analyzed by Burman and Hansbo (2006) [10], Badia and Codina (2009) [11], Becker and Braack (2001) [12], and Nafa and Wathen (2009) [13]. In this work we consider a coupled Stokes–Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous finite elements in the two parts, such as Taylor-Hood or MINI elements for the Stokes domain, and mixed elements of Raviart–Thomas elements type for the Darcy domain. Here, we discretize by standard equal-order finite elements enriched with bubbles functions and use local projection stabilization technique (LPS) to stabilize the method and control the fluctuation of the velocity divergence vector on the Darcy region. We also suggest a way to control the natural H(div) velocity.