Raviart-Thomas Elements Research Articles

Superconvergence analysis of an [formula omitted]-Galerkin mixed finite element method for nonlinear BBM equation

An H1-Galerkin mixed finite element method (MFEM) is developed for the Benjamin–Bona–Mahony (BBM) equation with the bilinear element and zero order Raviart–Thomas element pair (Q11∕Q10×Q01). Then based on the special interpolation properties of the above two elements and the mean-value technique, the supercloseness and superconvergence results of order O(h2) for the semi-discrete scheme and of order O(h2+(Δt)2) for the Crank–Nicolson fully-discrete scheme are deduced for the original variable u in H1-norm and the flux p→ in H(div,Ω)-norm, respectively. Finally, a numerical example is carried out to demonstrate the theoretical analysis.

A High Order Mixed-FEM for Diffusion Problems on Curved Domains

We propose and analyze a high order mixed finite element method for diffusion problems with Dirichlet boundary condition on a domain $$\Omega $$ with curved boundary $$\Gamma $$ . The method is based on approximating $$\Omega $$ by a polygonal subdomain $$\mathrm {D}_{h}$$ , with boundary $$\Gamma _h$$ , where a high order conforming Galerkin method is considered to compute the solution. To approximate the Dirichlet data on the computational boundary $$\Gamma _h$$ , we employ a transferring technique based on integrating the extrapolated discrete gradient along segments joining $$\Gamma _h$$ and $$\Gamma $$ . Considering general finite dimensional subspaces we prove that the resulting Galerkin scheme, which is $${\mathbf {H}}(\mathrm {div}\,; \mathrm {D}_{h})$$ -conforming, is well-posed provided suitable hypotheses on the aforementioned subspaces and integration segments. A feasible choice of discrete spaces is given by Raviart–Thomas elements of order $$k\ge 0$$ for the vectorial variable and discontinuous polynomials of degree k for the scalar variable, yielding optimal convergence if the distance between $$\Gamma _h$$ and $$\Gamma $$ is at most of the order of the meshsize h. We also approximate the solution in $$\mathrm {D}_{h}^{c}\,{:}{=}\,\Omega \backslash \overline{\mathrm {D}_{h}}$$ and derive the corresponding error estimates. Numerical experiments illustrate the performance of the scheme and validate the theory.

Expanded mixed FEM with lowest order RT elements for nonlinear and nonlocal parabolic problems

In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Euler method, a complete discrete scheme is proposed and a variant of Brouwer’s fixed point theorem is used to derive an existence of a fully discrete solution. Further, a priori error estimates for the fully discrete scheme are established. Finally, numerical experiments are conducted to confirm our theoretical findings.

Flux reconstructions in the Lehmann–Goerisch method for lower bounds on eigenvalues

The standard application of the Lehmann–Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes σ̃i that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes σ̃i are usually computed by solving a global saddle point problem with mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes σ̃i of the same quality. The simplified problem is smaller, it is positive definite, and any H(div,Ω) conforming finite elements, such as Raviart–Thomas elements, can be used for its solution. In addition, these global problems can be split into a number of independent local problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions.

A finite element method for Maxwell polynomial chaos Debye model

‘In this paper, a finite element method is presented to approximate Maxwell–Polynomial Chaos(PC) Debye model in two spatial dimensions. The existence and uniqueness of the weak solutions are presented firstly according with the differential equations by using the Laplace transform. Then the property of energy decay with respect to the time is derived. Next, the lowest Nédélec–Raviart–Thomas element is chosen in spatial discrete scheme and the Crank–Nicolson scheme is employed in time discrete scheme. The stability of full-discrete scheme is explored before an error estimate of accuracy O(Δt2+h) is proved under the L2−norm. Numerical experiment is demonstrated for showing the correctness of the results.

On a vorticity-based formulation for reaction-diffusion-Brinkman systems

We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nedelec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

Convergence analysis of a BDF2 / mixed finite element discretization of a Darcy–Nernst–Planck–Poisson system

This paper presents an a priori error analysis of a fully discrete scheme for the numerical solution of the transient, nonlinear Darcy–Nernst–Planck–Poisson system. The scheme uses the second order backward difference formula (BDF2) in time and the mixed finite element method with Raviart–Thomas elements in space. In the first step, we show that the solution of the underlying weak continuous problem is also a solution of a third problem for which an existence result is already established. Thereby a stability estimate arises, which provides an L ∞ bound of the concentrations / masses of the system. This bound is used as a level for a cut-off operator that enables a proper formulation of the fully discrete scheme. The error analysis works without semi-discrete intermediate formulations and reveals convergence rates of optimal orders in time and space. Numerical simulations validate the theoretical results for lowest order finite element spaces in two dimensions.

RESIDUAL-BASED A POSTERIORI ERROR ESTIMATES FOR A CONFORMING FINITE ELEMENT DISCRETIZATION OF THE NAVIER-STOKES/DARCY COUPLED PROBLEM

We consider in this paper, a new a posteriori residual type error estimator of a conforming mixed finite element method for the coupling of fluid flow with porous media flow on isotropic meshes. 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. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. The posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. In addition, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

Decoupling mixed finite elements on hierarchical triangular grids for parabolic problems

In this paper, we propose a numerical method for the solution of time-dependent flow problems in mixed form. Such problems can be efficiently approximated on hierarchical grids, obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements. If these elements are considered as subdomains, we can formulate a non-overlapping domain decomposition method based on the lowest-order Raviart–Thomas elements, properly enhanced with Lagrange multipliers on the boundaries of each subdomain (excluding the Dirichlet edges). A suitable choice of mixed finite element spaces and quadrature rules yields a cell-centered scheme for the pressures with a local 10-point stencil. The resulting system of differential-algebraic equations is integrated in time by the Crank–Nicolson method, which is known to be a stiffly accurate scheme. As a result, we obtain independent subdomain linear systems that can be solved in parallel. The behavior of the algorithm is illustrated on a variety of numerical experiments.

High accuracy analysis of an [formula omitted]-Galerkin mixed finite element method for two-dimensional time fractional diffusion equations

In this paper, an H1-Galerkin mixed finite element approximate scheme is established for a class of two-dimensional time fractional diffusion equations by using the bilinear element, Raviart–Thomas element and L1 time stepping method, which is unconditionally stable and free of LBB condition. And then, without the classical Ritz projection, superclose results for the original variable u in H1-norm and the flux p→=∇u in H(div,Ω)-norm are derived by means of properties of the elements and L1 approximation. Furthermore, with the help of the interpolation postprocessing operator, the global superconvergence results for the original variable u in H1-norm are obtained. Finally, numerical simulations verify that the theoretical results are true on both regular meshes and anisotropic meshes.

Superconvergence analysis of an H1-Galerkin mixed finite element method for two-dimensional multi-term time fractional diffusion equations

ABSTRACTIn this paper, numerical approximation for two-dimensional (2D) multi-term time fractional diffusion equation is considered. By virtue of properties of bilinear element, Raviart–Thomas element and approximation, an -Galerkin mixed finite element fully discrete approximate scheme is established for 2D multi-term time fractional diffusion equation. And then, unconditionally stable of the approximate scheme is rigourously testified by dealing with fractional derivative skilfully. At the same time, superclose results for the original variable u in -norm and the flux in -norm are derived. Furthermore, the global superconvergence results for u in -norm are deduced by the interpolation postprocessing operator. Finally, numerical results demonstrate that the approximate scheme provides a valid and efficient way for solving 2D multi-term time fractional diffusion equation.

High accuracy analysis of the lowest order H1-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H1-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h2)/O(h2 + τ2) in H1-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.

Wave dispersion properties of compound finite elements

Mixed finite elements use different approximation spaces for different dependent variables. Certain classes of mixed finite elements, called compatible finite elements, have been shown to exhibit a number of desirable properties for a numerical weather prediction model. In two-dimensions the lowest order element of the Raviart–Thomas based mixed element is the finite element equivalent of the widely used C-grid staggering, which is known to possess good wave dispersion properties, at least for quadrilateral grids. It has recently been proposed that building compound elements from a number of triangular Raviart–Thomas sub-elements, such that both the primal and (implied) dual grid are constructed from the same sub-elements, would allow greater flexibility in the use of different advection schemes along with the ability to build arbitrary polygonal elements. Although the wave dispersion properties of the triangular sub-elements are well understood, those of the compound elements are unknown. It would be useful to know how they compare with the non-compound elements and what properties of the triangular sub-grid elements are inherited?Here a numerical dispersion analysis is presented for the linear shallow water equations in two dimensions discretised using the lowest order compound Raviart–Thomas finite elements on regular quadrilateral and hexagonal grids. It is found that, in comparison with the well known C-grid scheme, the compound elements exhibit a more isotropic dispersion relation, with a small over estimation of the frequency for short waves compared with the relatively large underestimation for the C-grid. On a quadrilateral grid the compound elements are found to differ from the non-compound Raviart–Thomas quadrilateral elements even for uniform elements, exhibiting the influence of the underlying sub-elements. This is shown to lead to small improvements in the accuracy of the dispersion relation: the compound quadrilateral element is slightly better for gravity waves but slightly worse for inertial waves than the standard lowest order Raviart–Thomas element.

A Raviart-Thomas mixed finite element scheme for the two-dimensional three-temperature heat conduction problems

Summary For the two-dimensional three-temperature radiative heat conduction problem appearing in the inertial confinement numerical stimulations, we choose the Freezing coefficient method to linearize the nonlinear equations, and initially apply the well-known mixed finite element scheme with the lowest order Raviart–Thomas element associated with the triangulation to the linearized equations, and obtain the convergence with one order with respect to the space direction for the temperature and flux function approximations, and design a simple but efficient algorithm for the discrete system. Three numerical examples are displayed. The former two verify theoretical results and show the super-convergence for temperature and flux functions at the barycenter of the element, which is helpful for solving the radiative heat conduction problems. The third validates the robustness of this scheme with small energy conservative error and one order convergence for the time discretization. Copyright © 2016 John Wiley & Sons, Ltd.

A Mortar Method Using Nonconforming and Mixed Finite Elements for the Coupled Stokes-Darcy Model

AbstractIn this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.

Robust fixed stress splitting for Biot’s equations in heterogeneous media

We study the iterative solution of coupled flow and geomechanics in heterogeneous porous media, modeled by a three-field formulation of the linearized Biot’s equations. We propose and analyze a variant of the widely used Fixed Stress Splitting method applied to heterogeneous media. As spatial discretization, we employ linear Galerkin finite elements for mechanics and mixed finite elements (lowest order Raviart–Thomas elements) for flow. Additionally, we use implicit Euler time discretization. The proposed scheme is shown to be globally convergent with optimal theoretical convergence rates. The convergence is rigorously shown in energy norms employing a new technique. Furthermore, numerical results demonstrate robust iteration counts with respect to the full range of Lamé parameters for homogeneous and heterogeneous media. Being in accordance with the theoretical results, the iteration count is hardly influenced by the degree of heterogeneities.

Dual-mixed finite elements for the three-field Stokes model as a finite volume method on staggered grids

In this paper, a new three-field weak formulation for Stokes problems is developed, and from this, a dual-mixed finite element method is proposed on a rectangular mesh. In the proposed mixed methods, the components of stress tensor are approximated by piecewise constant functions or Q1 functions, while the velocity and pressure are discretized by the lowest-order Raviart-Thomas element and the piecewise constant functions, respectively. Using quadrature rules, we demonstrate that this scheme can be reduced into a finite volume method on staggered grid, which is extensively used in computational fluid mechanics and engineering.

A posteriori error analysis of a fully-mixed formulation for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity

In this paper we consider an augmented fully-mixed variational formulation that has been recently proposed for the coupling of the Navier–Stokes equations (with nonlinear viscosity) and the linear Darcy model, and derive a reliable and efficient residual-based a posteriori error estimator for the associated mixed finite element scheme. The finite element subspaces employed are piecewise constants, Raviart–Thomas elements of lowest order, continuous piecewise linear elements, and piecewise constants for the strain, Cauchy stress, velocity, and vorticity in the fluid, respectively, whereas Raviart–Thomas elements of lowest order for the velocity, piecewise constants for the pressure, and continuous piecewise linear elements for the traces, are considered in the porous medium. The proof of reliability of the estimator relies on a global inf–sup condition, suitable Helmholtz decompositions in the fluid and the porous medium, the local approximation properties of the Clément and Raviart–Thomas operators, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

A mortar finite volume method for a fractured model in porous media

In this paper, we consider a mortar finite volume method for a fractured model of flow in porous media. In this model, the permeability coefficients are variable between the fracture and the surrounding porous media. A finite volume method based on Raviart–Thomas elements combined with the mortar technique of domain decomposition is presented, in which sub-domains are triangulated independently and the meshes do not match at interfaces. The great advantage of the method is avoiding solving the saddle-point problem, since the numerical scheme is just related to the pressure p, and the velocity u can be expressed by p. We also prove error estimates of order h on the discrete H1 norm between the exact solution p and the mortar finite volume solution P and the (L2)2 norm between u and U. Finally, numerical experiments have been performed to show the consistency of the convergence rates with the theoretical analysis.

Compatible-strain mixed finite element methods for 2D compressible nonlinear elasticity

In this paper, using the Hilbert complexes of nonlinear elasticity, the approximation theory for Hilbert complexes, and the finite element exterior calculus, we introduce a new class of mixed finite element methods for 2D nonlinear elasticity–compatible-strain mixed finite element methods (CSFEM). We consider a Hu–Washizu-type mixed formulation and choose the displacement, the displacement gradient, and the first Piola–Kirchhoff stress tensor as independent unknowns. We use the underlying spaces of the Hilbert complexes as the solution and test spaces. We discretize the Hilbert complexes and introduce a new class of mixed finite element methods for nonlinear elasticity by using the underlying finite element spaces of the discrete Hilbert complexes. This automatically enforces the trial spaces of the displacement gradient to satisfy the classical Hadamard jump condition, which is a necessary condition for the compatibility of non-smooth displacement gradients. The underlying finite element spaces of CSFEMs are the tensorial analogues of the standard Nédélec and Raviart–Thomas elements of vector fields. These spaces respect the global topologies of the domains in the sense that they can reproduce certain topological properties of the bodies regardless of the refinement level of meshes. By solving several numerical examples, we demonstrate that CSFEMs have a good performance for bending problems, in the near incompressible regime, and for bodies with complex geometries. CSFEMs are capable of accurately approximating stresses and perform well in problems that standard enhanced strain methods suffer from the hourglass instability. Moreover, CSFEMs provide a convenient framework for modeling inhomogeneities.