Lowest Order Raviart-Thomas Elements Research Articles

Numerical Algorithms for Divergence-Free Velocity Applications

This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart–Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier–Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation.

Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method.

A finite volume method for a coupled fracture model with matching and nonmatching grids

In this paper, we propose and analyze a 4-point finite volume method for a fracture model coupling one-dimensional equations in the fracture with two-dimensional equations in surrounding domains. The pressure is approximated in the piecewise constant spaces, whereas the velocity is calculated by the lowest order Raviart-Thomas elements and piecewise constants in matrix and fracture, respectively. Optimal order error estimates are proved on nonuniform triangular meshes for both the pressure and velocity. Beside, we extend the 4-point finite volume method to nonmatching grids between the fracture and matrix without loss of any accuracy. Numerical experiments on matching and nonmatching meshes are tested for models with higher, lower and anisotropic fracture permeability, and results confirm our theoretical analysis.

A discontinuous finite volume method for a coupled fracture model

In this paper, we present a discontinuous finite volume method for a hybrid-dimensional coupled fracture model on triangular meshes. In matrix and fracture, the pressure is approximated by the discontinuous and continuous piecewise linear elements, and in order to retain local conservation property, the velocity is sought in the discontinuous lowest-order Raviart–Thomas element space and piecewise linear element space. The proposed numerical scheme is only associated with the pressure, so we avoid solving an indefinite saddle-point problem. In addition, the discontinuous method eliminates the continuity of approximate functions across the interelement boundary, so it has the advantage of high localizability and easy handling of complicated geometries. Optimal order error estimates for the pressure and velocity are proved, and numerical experiments are carried out to confirm our theoretical analysis.

Aspects of finite element formulations for the coupled problem of poroelasticity based on a canonical minimization principle

This work presents a new finite element treatment of the coupled problem of Darcy–Biot-type fluid transport in porous media undergoing large deformations, that is free from any stabilization techniques. The formulation bases on an incremental two-field minimization principle that is constrained by the equation of continuity for the fluid mass content and determines at a given state the deformation and the fluid mass flux vector. The big advantage of the minimization formulation over classical saddle point principles of poroelasticity is the omission of the inf-sup condition—a condition that makes the construction of stable and computationally efficient finite element formulations difficult. Due to the $$H(\hbox {Div}, {\mathcal B}_0)$$ variational structure of the minimization principle on the fluid side, lowest order Raviart–Thomas elements are used for the conforming approximation of the fluid mass flux. Furthermore, a standard nodal-based element using bilinear interpolation for both fields combined with a reduced numerical integration of the (volumetric) coupling term is analyzed and used for the solution of the minimization principle. Representative numerical examples demonstrate the performance of the proposed finite element designs of the minimization principle and clearly underline advantages over finite element formulations of the classical two-field saddle point principle formulated in deformation and fluid potential.

Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity

We analyze a fully discrete numerical scheme for the model describing two-phase immiscible flow in porous media with dynamic effects in the capillary pressure. We employ the Euler implicit method for the time discretization. The spatial discretization is based on the mixed finite element method (MFEM). Specifically, the lowest order Raviart–Thomas elements are applied. In this paper, the error estimates for the saturation, fluxes and phase pressures in L ∞ ( 0 , T ; L 2 ( Ω ) ) are derived for the temporal and spatial discretization to show the convergence of the scheme. Finally, we present some numerical results to support the theoretical findings.

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.

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.

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.

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 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).

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.

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.

A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients

We aim at proposing novel stable finite element methods for the mixed Darcy equation with heterogeneous coefficients within a space splitting framework. We start from the primal hybrid formulation of the elliptic model for the pressure. Localization of this infinite-dimensional problem leads to element-level boundary value problems which embed multiscale and high-contrast features in a natural way, with Neumann boundary conditions driven by the Lagrange multipliers. Such a procedure leads to methods involving the space of piecewise constants for the pressure together with a discretization of the fluxes. Choosing (arbitrarily) polynomial interpolations, the lowest-order Raviart–Thomas element as well as some recent multiscale methods are recovered. In addition, the methods assure local mass conservation and can be interpreted as stabilized primal hybrid methods. Extensive numerical validation attests to the accuracy of the new methods on academic and more realistic problems with rough coefficients.

A divergence-free velocity reconstruction for incompressible flows

In incompressible flows with vanishing normal velocities at the boundary, irrotational forces in the momentum equations should be balanced completely by the pressure gradient. Unfortunately, nearly all available discretizations for incompressible flows violate this property. The origin of the problem is that discrete velocities are usually not divergence-free. Hence, the use of divergence-free velocity reconstructions is proposed wherever an L2 scalar product appears in the discrete variational formulation. The approach is illustrated and applied to a nonconforming MAC-like discretization for unstructured Delaunay grids. It is numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart–Thomas element increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.

Hybrid finite element analysis of fluid-structure systems with coupling on curved interfaces

This paper deals with a finite element approximation of the vibration modes of fluid–structure systems coupled on curved interfaces. It is based on a displacement formulation for both the fluid and the solid. Lowest order Raviart–Thomas elements are used for the fluid and standard continuous linear elements for the solid. Compatibility conditions are weakly imposed at a polygonal approximation of the fluid–solid interface by means of a Lagrange multiplier. Convergence, nonexistence of spurious or circulation nonzero frequency modes and optimal order error estimates for eigenfunctions/eigenvalues are proved. To do this, we use recent results about spectral approximations for noncompact operators to nonconforming hybrid finite element methods. The validity of a discrete compactness property for the discrete spaces considered here is also discussed.

A QUASI-VARIATIONAL INEQUALITY PROBLEM IN SUPERCONDUCTIVITY

We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical finite element method based on the lowest order Raviart–Thomas element, which yields approximations to both the primal and dual variables (the magnetic and electric fields). We prove the subsequence convergence of this approximation, and hence prove the existence of a solution to both the dual and primal formulations, for strictly star-shaped domains. The effectiveness of the approximation is illustrated by numerical examples with and without this domain restriction.