Galerkin Mixed Finite Element Research Articles

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

A locking-free numerical method for the quasi-static linear thermo-poroelasticity model

In this paper, we propose a locking-free numerical method to solve the quasi-static linear thermo-poroelasticity model, which exhibits two types of locking phenomena: Poisson locking and nonphysical oscillations. By analyzing the regularity of solution of the original model, we find that the Poisson locking is caused by divu≈0 as λ→+∞. Moreover, when discretizing the diffusion equations using backward Euler method for time, one can see that divu1≈0 for some special parameters. If we directly use the continuous Galerkin mixed finite element method (FEM) to solve the original model, the pressure and temperature fields exhibit numerical oscillations at early times. To overcome these two locking phenomena, we introduce a new variable ξ=αp+βT−λdivu to reformulate the original problem into a new one. This new problem includes a built-in mechanism to maintain stability for the continuous Galerkin mixed FEM. We further prove the existence and uniqueness of the weak solution by using the standard Galerkin method in conjunction with regularity estimates. We also design a fully discrete time-stepping scheme that employs mixed FEM with P2−P1−P1−P1 element pairs for the space variables and the backward Euler method for the time variable. Optimal convergence is demonstrated in both space and time. Finally, numerical examples are provided to demonstrate the optimal convergence rates of the variables and the robustness of the proposed method with respect to ν, and to verify the absence of the locking phenomenon.

A Weak Galerkin Mixed Finite Element Method for Linear Elasticity Without Enforced Symmetry

A Weak Galerkin Mixed Finite Element Method for Linear Elasticity Without Enforced Symmetry

Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.

Analysis of aWeak Galerkin Mixed Formulation for Maxwell’s Equations

In this paper we introduce and analyse a mixed weak Galekin finite element method for the Maxwell equations in the primary electric field-Lagrange multiplier. Our weak Galerkin method is equipped with stable finite elements composed of habitual polynomials of degree k for the electric field and polynomials of degree k + 1 for the Lagrange multiplier. Optimal order error estimations for the proposed weak Galerkin mixed finite element formulation are demonstrated and are confirmed numerically on a two dimensional bounded domain.

Galerkin mixed finite element method for parabolic p-biharmonic equation with memory term

Galerkin mixed finite element method for parabolic p-biharmonic equation with memory term

Optimal Analysis of Non-Uniform Galerkin-Mixed Finite Element Approximations to the Ginzburg–Landau Equations in Superconductivity

This paper is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg–Landau equations. The method is based on a commonly-used nonuniform approximation, in which a linear Lagrange element, the lowest-order Nédélec edge element, and the Raviart–Thomas face element are used for the order parameter , the magnetic field , and the magnetic potential , respectively. This mixed method has been widely used in practical simulations due to its low cost and ease of implementation. In the Ginzburg–Landau model, the order parameter is the most important variable, which indicates the state of the superconductor. An important feature of the method is the inconsistency of the approximation orders. A crucial question is how the first-order approximation of influences the accuracy of . The main purpose of this paper is to establish the second-order accuracy for the order parameter in a spatial direction, although the accuracy for is first order only. Previous analysis only gave the first-order convergence for all three variables due to certain artificial pollution involved in the analysis. Our analysis is based on a nonstandard quasi-projection for and the corresponding more precise estimates, including the -norm. With the quasi-projection, we prove that the lower-order approximation to does not pollute the accuracy of . Our numerical experiments confirm the optimal convergence of . The approach can be extended to many other multiphysics models.

An $$H^{1}-$$Galerkin mixed finite element method for rosenau equation

An $$H^{1}-$$Galerkin mixed finite element method for rosenau equation

Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations

<abstract><p>In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by $ H^1 $-Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and $ L1 $ scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples.</p></abstract>

Superclose analysis of $H^1$-Galerkin mixed finite element methods combined with two-grid scheme for semilinear parabolic equations

Superclose analysis of $H^1$-Galerkin mixed finite element methods combined with two-grid scheme for semilinear parabolic equations

An [formula omitted] weak Galerkin mixed finite element method for Sobolev equation

In this paper, we present a new H1 weak Galerkin mixed finite element method for the Sobolev equation which includes the exact solution u and the intermediate solution p. In the H1 weak Galerkin method, we adopt the discontinuous finite elements Pk/Pk for the approximation solution pair (uh,ph) on finite element partitions consisting of arbitrary shape of polygons which are WG shape regular. We give the semi-discrete and full-discrete formulations which are proven to be stable and parameter-free and possess the optimal error estimates. Numerical experiments show the efficiency of our methods.

Optimal error estimates of a lowest-order Galerkin-mixed FEM for the thermoviscoelastic Joule heating equations

This paper is concerned with the optimal error estimates of a classical Galerkin-mixed finite element method (FEM) for the thermoviscoelastic Joule heating equations, which couples the temperature, the electric potential and the deformation of a thermoviscoelastic body. The method is based on a popular combination of the lowest-order Raviart-Thomas mixed approximation for the electric potential/field (ϕ,θ) and the linear Lagrange approximation for the temperature u and the deformation b. By using the temporal-spatial error splitting techniques, we prove that the method produces the optimal second-order accuracy O(h2) for u and b in the spatial direction, and the accuracy O(h) for the potential/field without any restriction on the time step size. Moreover, a simple single-step recovery method is introduced to improve the accuracy for the electric potential/field to O(h2). Numerical results are provided to confirm our theoretical analysis and show clearly that no time-step condition is needed.

A Weak Galerkin Mixed Finite Element Method for Second Order Elliptic Equations on 2D Curved Domains

This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary condition in this case. It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretization with polynomial order $\alpha>1$. The purpose of this paper is two-fold. First, we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains, which exhibits an $O(h^{1/2})$ convergence for all $\alpha\ge 1$. It is a little surprising to see that even the lowest-order WG-MFEM ($\alpha=1$) experiences a loss of accuracy. This is different from known results for the finite element method (FEM) or the mixed FEM, and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain. Second, we propose a remedy to bring the approximation rate back to optimal by employing two techniques. One is a specially designed boundary correction technique. The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes, which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements. Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all $\alpha$. Numerical results further confirm this conclusion.

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations

A Posteriori Error Estimates of Edge Residual-Type of Weak Galerkin Mixed FEM Solving Second-Order Elliptic Problems on Polytopal Mesh

In this paper, we present a posteriori error estimator of edge residual-type Weak Galerkin mixed finite element method (WG-MFEM) solving second-order elliptic problems, where two different ways of a posteriori error estimator are presented, both of which hold on polygonal mesh. The two analyses are based on the standard FEM method and continuous local mass conservation property, respectively. In order to prove the reliability holding on polygonal meshes, the a posteriori error estimator is disposed by the post-processed numerical scalar solution. Compared with some other methods, the proposed a posteriori error estimator does not contain the post-processed solution and the efficiency can be obtained automatically. Numerical experiments are conducted to confirm the efficiency of this estimator.

The Galerkin-Implicit Method for Solving Nonlinear Variable Coefficients Hyperbolic Boundary Value Problem

This paper has the interest of finding the approximate solution (APPS) of a nonlinear variable coefficients hyperbolic boundary value problem (NOLVCHBVP). The given boundary value problem is written in its discrete weak form (WEFM) and proved have a unique solution, which is obtained via the mixed Galerkin finite element with implicit method that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector techniques (PT and CT, respectively) are proved at first convergence and then are used to transform the obtained GNAS to a linear GLAS . Then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. Some illustrative examples are used, where the results are given by figures that show the efficiency and accuracy for the method.

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation. A discrete inf-sup condition is proved and the optimal error estimates are also derived. Numerical experiments validate the theoretical analysis.

The Galerkin-Implicit Methods for Solving Nonlinear Hyperbolic Boundary Value Problem

This paper is concerned with finding the approximation solution (APPS) of a certain type of nonlinear hyperbolic boundary value problem (NOLHYBVP). The given BVP is written in its discrete (DI) weak form (WEF), and is proved that it has a unique APPS, which is obtained via the mixed Galerkin finite element method (GFE) with implicit method (MGFEIM) that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector technique (PT and CT) are proved convergent and are used to transform the obtained GNAS to linear (GLAS ), then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. The results are given by figures and shown the efficiency and accuracy for the method

Mixed Implicit Galerkin – Frank Wolf, Gradient and Gradient Projection Methods for Solving Classical Optimal Control Problem Governed by Variable Coefficients, Linear Hyperbolic, Boundary Value Problem

This paper deals with testing a numerical solution for the discrete classical optimal control problem governed by a linear hyperbolic boundary value problem with variable coefficients. When the discrete classical control is fixed, the proof of the existence and uniqueness theorem for the discrete solution of the discrete weak form is achieved. The existence theorem for the discrete classical optimal control and the necessary conditions for optimality of the problem are proved under suitable assumptions. The discrete classical optimal control problem (DCOCP) is solved by using the mixed Galerkin finite element method to find the solution of the discrete weak form (discrete state). Also, it is used to find the solution for the discrete adjoint weak form (discrete adjoint) with the Gradient Projection method (GPM) , the Gradient method (GM), or the Frank Wolfe method (FWM) to the DCOCP. Within each of these three methods, the Armijo step option (ARSO) or the optimal step option (OPSO) is used to improve (to accelerate the step) the solution of the discrete classical control problem. Finally, some illustrative numerical examples for the considered discrete control problem are provided. The results show that the GPM with ARSO method is better than GM or FWM with ARSO methods. On the other hand, the results show that the GPM and GM with OPSO methods are better than the FWM with the OPSO method.

A new discontinuous Galerkin mixed finite element method for compressible miscible displacement problem

A new combined discontinuous Galerkin method is proposed for compressible miscible displacement problem in porous media. Here, a splitting positive definite mixed finite element method is used for the pressure and Darcy velocity, while an interior penalty discontinuous Galerkin (IPDG) method is used for the transport equation. The stability and convergence of this algorithm are considered, and the optimal a priori error estimate in l∞(L2) for velocity, pressure and concentration are given. Finally we provide some numerical results to confirm our theoretical analysis, and simulate compressible fluid flows through homogeneous and isotropic porous media.