Low-order Mixed Finite Element Research Articles

New low-order mixed finite element methods for linear elasticity

New low-order mixed finite element methods for linear elasticity

Stabilized low-order mixed finite element methods for a Navier-Stokes hemivariational inequality

Stabilized low-order mixed finite element methods for a Navier-Stokes hemivariational inequality

Two‐level multiscale enrichment finite volume method based on the Newton iteration for the stationary incompressible magnetohydrodynamics flow

In this paper, we propose a stabilized finite volume method based on the Newton's type iteration for the steady incompressible magnetohydrodynamics (MHD) problem. In order to reduce the computational complexity, the lowest order mixed finite element pair ( ‐ ‐ ) is adopted to approximate the velocity, pressure and magnetic fields, and the multiscale enrichment method is introduced to overcome the restriction of discrete inf‐sup (LBB) condition. We firstly solve the incompressible MHD equations by the Newton iterations on a coarse mesh with the mesh size , stability and convergence results of numerical solutions are provided. Secondly, the combination of the two‐level method with the Newton iteration is used to approximate the considered problem, a coupled nonlinear problem is solved on the coarse mesh , then the Stokes and Maxwell equations are considered on a fine mesh with the mesh size , the uniform stability and optimal error estimates of two‐level Newton iterative method are provided. Theoretical findings show that the two‐level method has the same order as the one‐level method in ‐norm as long as the mesh sizes satisfy . However, the two‐level method involves much less work than the one‐level method. Finally, some numerical examples are presented to demonstrate the effectiveness of the considered numerical schemes.

The optimal order convergence for the lowest order mixed finite element method of the biharmonic eigenvalue problem

In this paper, we present the theoretical analysis of the optimal order convergence for the piecewise linear and continuous finite element method based on the Ciarlet–Raviart mixed formulation of the biharmonic eigenvalue problem associated to the clamped boundary condition. As far as we know, only the convergence of the equal order linear Ciarlet–Raviart finite element method for the eigenvalue problem has been established on convex domains. The aim to this work is to derive the convergence under the minimum regularity requirement and prove an improved convergence rate for the approximate eigenvalues. We introduce the corresponding solution spaces naturally attached to the continuous and discrete problems and prove the spectral approximation and error estimate of the discrete scheme. Some numerical examples are shown for the validation of the theoretical proof.

Nearly-constrained transversely isotropic linear elasticity: energetically consistent anisotropic deformation modes for mixed finite element formulations

Strong anisotropies and/or near-incompressibility properties introduce internal constraints in material deformation. Numerical simulations comprising such a constrained behaviour show an overstiff structural response, referred to as element locking. Implementations based on mixed variational methods can heal locking but available solutions in the state-of-the-art are still non-optimal for anisotropic materials. This paper addresses this issue, by proposing a novel decomposition of anisotropic deformation modes on the basis of kinematic and energy requirements. Theoretical results exploit the Walpole’s formalism. The proposed kinematic split allows to introduce a new class of variational principles, referred to as energetically decoupled, for nearly-constrained transversely isotropic materials in linear elasticity. Low-order mixed finite element models are thus derived for treating near-inextensibility and/or near-incompressibility. Two-dimensional benchmark tests reproducing pure-bending and Cook’s membrane problems are conducted. Numerical results show that the accuracy of energetically decoupled formulations is high and robust with respect to variations of material properties, while the accuracy of non-energetically decoupled formulations is more sensitive.

A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem

The goal of this paper is to develop numerical methods computing a few smallest elastic interior transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. The problem is challenging since it is nonlinear, nonselfadjoint, and of fourth order. In this paper, we construct a lowest order mixed finite element method which is close to the Ciarlet-Raviart mixed finite element method. This scheme is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. Due to the nonselfadjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixed method which does not fit into the framework of classical theoretical analysis. In stead, we obtain the convergence analysis based on the spectral approximation theory of compact operators. Numerical examples are presented to verify the theory. Both real and complex eigenvalues can be obtained.

A Finite Element/Operator-Splitting Method for the Numerical Solution of the Three Dimensional Monge–Ampère Equation

In the present article we extend to the three-dimensional elliptic Monge–Ampere equation the method discussed in Glowinski et al. (J Sci Comput 79:1–47, 2019) for the numerical solution of its two-dimensional variant. As in Glowinski et al. (2019) we take advantage of an equivalent divergence formulation of the Monge–Ampere equation, involving the cofactor matrix of the Hessian of the solution. We associate with the above divergence formulation an initial value problem, well suited to time discretization by operator splitting and space approximation by low order mixed finite element methods. An important ingredient of our methodology is forcing the positive semi-definiteness of the approximate Hessian by a hard thresholding eigenvalue projection. The resulting method is robust and easy to implement. It can handle problems with smooth and non-smooth solutions on domains with curved boundary. Using piecewise affine approximations for the solution and its six second-order derivatives, one can achieve second-order convergence rates for problems with smooth solutions.

Low-order Raviart–Thomas approximations of axisymmetric Darcy flow

Abstract In this paper, we study the lowest-order mixed finite element method for the axisymmetric Darcy problem using Raviart–Thomas elements. In contrast to the Cartesian setting, the method is non-conforming in the sense that the discretely divergence-free functions are not solenoidal. We derive several estimates that measure the inconsistency of the method and derive error estimates of the discrete pressure and velocity solutions. We show that if the domain is convex, then the errors converge with optimal order modulo logarithmic terms. Numerical experiments are presented, and they indicate that the estimates are sharp.

An approximate Jacobian nonlinear solver for multiphase flow and transport

We present an approximate Jacobian approach for solving nonlinear, multiphase flow and transport problems in porous media. A backward Euler time discretization scheme is used prior to spatial discretization with a lowest order mixed finite element method (MFEM). This results in a fully implicit nonlinear algebraic system of equations. Conventionally, an exact Jacobian construction is employed during the Newton linearization to obtain a linear system of equations after spatial and temporal discretization. This fully coupled, monolithic linear system, usually in pressure and saturation (or concentration) unknowns, requires specialized preconditioners such as constrained pressure residual (CPR) or two stage preconditioner. These preconditioners operate on the linear system to decouple pressure and saturation (or concentration) degrees of freedom (DOF) in order to use existing linear solvers for positive definite (PD) matrices such as GMRES or AMG, to name a few. In this work, we present an alternative to two-stage preconditioning (or CPR) for solving the aforementioned monolithic system after Newton linearization. This approach relies upon a decoupling approximation for the pressure-saturation (or concentration) block sub-matrices, during Newton linearization, to obtain block diagonal sub-matrices. The resulting linear system is easily reduced, trivially eliminating these diagonal sub-matrices, to obtain a system in pressure DOF circumventing the need for specialized preconditioners. Further, the linear system has lesser DOF owing to the elimination of saturation (or concentration) unknowns. This nonlinear solver is demonstrated to be as accurate as the exact Jacobian approach, measured in terms of convergence of nonlinear residual to a desired tolerance for both methods. Our numerical results indicate a consistent computational speedup by a factor of approximately 1.32 to 4.0 for the two-phase flow model formulation under consideration. This is related to the DOF of the linear systems for the approximate and exact Jacobian approaches. For multicomponent flow and transport, this speedup is expected to be directly proportional to the number of concentration degrees of freedom. A number of field scale numerical simulations are also presented to demonstrate the efficacy of this approach for realistic problems.

An Alternating Direction Method of Multipliers for the Numerical Solution of a Fully Nonlinear Partial Differential Equation Involving the Jacobian Determinant

We consider the Dirichlet problem for a partial differential equation involving the Jacobian determinant in two dimensions of space. The problem consists in finding a vector-valued function such that the determinant of its gradient is given pointwise in a bounded domain, together with essential boundary conditions. This problem was initially considered in Dacorogna and Moser [Ann. Inst. H. Poincare Anal. Non Lineaire, 7 (1990), pp. 1--26], and several theoretical generalizations have been derived since. In this work, we design a numerical algorithm for the approximation of the solution of such a problem for various kinds of boundary data. The proposed method relies on an augmented Lagrangian algorithm with biharmonic regularization, and low order mixed finite element approximations. An iterative method allows us to decouple the nonlinearity and the differential operators. Numerical experiments show the capabilities of the method for benchmarks and then for more demanding test problems. (An erratum has bee...

Global Superconvergence of the Lowest-Order Mixed Finite Element on Mildly Structured Meshes

In this paper, we develop global superconvergence estimates for the lowest order Raviart--Thomas mixed finite element method for second order elliptic equations with general boundary conditions on triangular meshes, where most pairs of adjacent triangles form approximate parallelograms. In particular, we prove the $L^{2}$-distance between the numerical solution and canonical interpolant for the vector variable is of order $1+\rho$, where $\rho\in(0,1]$ is dependent on the mesh structure. By a cheap local postprocessing operator $G_{h}$, we prove the $L^{2}$-distance between the exact solution and the postprocessed numerical solution for the vector variable is of order $1+\rho$. As a byproduct, we also obtain the superconvergence estimate for Crouzeix--Raviart nonconforming finite elements on triangular meshes of the same type.

Fast auxiliary space preconditioners for linear elasticity in mixed form

A block-diagonal preconditioner with the minimal residual method and an approximate block-factorization preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods for linear elasticity. They are based on a new stability result for the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while for the displacement it is spectral equivalent to the Schur complement. A fast auxiliary space preconditioner based on the H 1 H^1 -conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lamé constant and the mesh size. Numerical examples are presented to support theoretical results. As byproducts, a new stabilized low order mixed finite element method is proposed and analyzed and superconvergence results for the Hu-Zhang element are obtained.

Low-order mixed finite element analysis of progressive failure in pressure-dependent materials within the framework of the Cosserat continuum

PurposeThis paper aims to develop a finite element analysis strategy, which is suitable for the analysis of progressive failure that occurs in pressure-dependent materials in practical engineering problems.Design/methodology/approachThe numerical difficulties stemming from the strain-softening behaviour of the frictional material, which is represented by a non-associated Drucker–Prager material model, is tackled using the Cosserat continuum theory, while the mixed finite element formulation based on Hu–Washizu variational principle is adopted to allow the utilization of low-order finite elements.FindingsThe effectiveness and robustness of the low-order finite element are verified, and the simulation for a real-world landslide which occurred at the upstream side of Carsington embankment in Derbyshire reconfirms the advantages of the developed elastoplastic Cosserat continuum scheme in capturing the entire progressive failure process when the strain-softening and the non-associated plastic law are involved.Originality/valueThe permit of using low-order finite elements is of great importance to enhance computational efficiency for analysing large-scale engineering problems. The case study reconfirms the advantages of the developed elastoplastic Cosserat continuum scheme in capturing the entire progressive failure process when the strain-softening and the non-associated plastic law are involved.

Decoupled stabilized finite element methods for the Boussinesq equations with temperature-dependent coefficients

In this paper, the decoupled stabilized finite element methods are developed for the Boussinesq equations with temperature dependent coefficients. In our numerical schemes, the original problem is decoupled into the linearized Navier-Stokes equations and a parabolic problem, the low order mixed finite element pairs are used to approximate the spatial spaces and the backward Euler scheme is adopted to treat the time terms. Furthermore, the linear terms are dealt with the implicit scheme while the nonlinear terms are treated by the semi-implicit scheme, then, a lot of storage and computational cost are saved. The advantages for our numerical schemes are parameter-free, unconditionally stable, and constant matrix at each time level. Finally, some numerical tests are presented to verify the theoretical results of the developed numerical methods, and show that our schemes not only keep good accuracy but also save a lot of computational times.

Stabilization of low-order mixed finite elements for the plane elasticity equations

In this paper, we consider the mixed finite element method of the plane elasticity equations based on the Hellinger–Reissner variational principle. Low order mixed finite element spaces are used to approximate the stress and the displacement. Based on the local polynomial projection stabilization method, we present a stabilization scheme for the pairs to overcome the lack of the inf–supcondition. The stability is proved and the error estimate is derived. At last, two numerical examples are implemented to test the stability and effectiveness of the proposed method. These pairs are very convenient in the numerical implementation for the mixed form of the plane elasticity equations.

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method (FEM) for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q1rot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H1-norm and the pressure in the L2-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.

On enforcing maximum principles and achieving element-wise species balance for advection–diffusion–reaction equations under the finite element method

These figures show the fate of the product in a transient transport-controlled bimolecular reaction under vortex-stirred mixing. The left figure is obtained using a popular numerical formulation, which violates the non-negative constraint. The right figure is based on the proposed computational framework. These figures clearly illustrate the main contribution of this paper: The proposed computational framework produces physically meaningful results for advective–diffusive–reactive systems, which is not the case with many popular formulations. We present a robust computational framework for advective–diffusive–reactive systems that satisfies maximum principles, the non-negative constraint, and element-wise species balance property. The proposed methodology is valid on general computational grids, can handle heterogeneous anisotropic media, and provides accurate numerical solutions even for very high Péclet numbers. The significant contribution of this paper is to incorporate advection (which makes the spatial part of the differential operator non-self-adjoint) into the non-negative computational framework, and overcome numerical challenges associated with advection. We employ low-order mixed finite element formulations based on least-squares formalism, and enforce explicit constraints on the discrete problem to meet the desired properties. The resulting constrained discrete problem belongs to convex quadratic programming for which a unique solution exists. Maximum principles and the non-negative constraint give rise to bound constraints while element-wise species balance gives rise to equality constraints. The resulting convex quadratic programming problems are solved using an interior-point algorithm. Several numerical results pertaining to advection-dominated problems are presented to illustrate the robustness, convergence, and the overall performance of the proposed computational framework.

Stabilized low order finite elements for Stokes equations with damping

In this paper, we consider the stationary Stokes equations with damping. Low order mixed finite element spaces are used to approximate the velocity and the pressure, and a local pressure projection stabilization method is used for the pairs to overcome the lack of the inf-sup condition. The stability of this method is proved, and the optimal order error estimates are derived by some nonlinear analysis techniques. At last, two numerical examples are implemented to test the stability and effectiveness of the proposed method.

On the enhancement of low-order mixed finite element methods for the large deformation analysis of diffusion in solids

On the enhancement of low-order mixed finite element methods for the large deformation analysis of diffusion in solids

HIERARCHICAL ERROR ESTIMATORS FOR LOWEST-ORDER MIXED FINITE ELEMENT METHODS

In this work we study two a posteriori error estimators of hierarchical type for lowest-order mixed finite element methods.One estimator is computed by solving a global defect problem based on the splitting of the lowest-order Brezzi--Douglas--Marini space, and the other estimator is locally computable by applying the standard localization to the first estimator. We establish the reliability and efficiency of both estimators by comparing them with the standard residual estimator. In addition, it is shown that the error estimator based on the global defect problem is asymptotically exact under suitable conditions.