Galerkin Mixed Finite Element Method Research Articles

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.

Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.</p></abstract>

Superconvergence analysis of an [formula omitted]-Galerkin mixed FEM for Klein–Gordon–Zakharov equations with power law nonlinearity

This paper aims to study an H1-Galerkin mixed finite element method (FEM) for Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity. The bilinear element and zero-order Raviart–Thomas element (Q11/Q10×Q01) are taken as approximation spaces for the original variables p, ϕ and the auxiliary variables u=∇p and ψ=∇ϕ, respectively. The supercloseness and superconvergence estimates of order O(h2+τ2) are presented for p and ϕ in the H1 norm and u and ψ in the L2 norm, which improve the known results that the original variable ϕ can only reach optimal error estimate shown in numerical experiments without theoretical analysis. Here h is the subdivision parameter and τ is the time step. Finally, some numerical results are provided to support the theoretical analysis.

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

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.

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.

Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

In this article, some high-order time discrete schemes with an H 1 -Galerkin mixed finite element (MFE) method are studied to numerically solve a nonlinear distributed-order sub-diffusion model. Among the considered techniques, the interpolation approximation combined with second-order σ schemes in time is used to approximate the distributed order derivative. The stability and convergence of the scheme are discussed. Some numerical examples are provided to indicate the feasibility and efficiency of our schemes.

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

Weak Galerkin mixed finite element methods for parabolic equations with memory

We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both \begin{document}$ |\|·|\| $\end{document} and \begin{document}$ L^2 $\end{document} norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces \begin{document}$ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $\end{document} for \begin{document}$ k = 0 $\end{document} and the numerical convergence rates confirm the theoretical results.

A fourth‐order H1‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

A H1‐Galerkin mixed finite element method is applied to the Kuramoto–Sivashinsky equation by using a splitting technique, which results in a coupled system. The method described in this article may also be considered as a Petrov–Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since the second derivative of a cubic spline is a linear spline. Optimal‐order error estimates are obtained without any restriction on the mesh for both semi‐discrete and fully discrete schemes. The advantage of this method over that presented in Manickam et al., Comput. Math. Appl. vol. 35(6) (1998) pp. 5–25; for the same problem is that the size (i.e., (n + 1) × (n + 1)) of each resulting linear system is less than half of the size of the linear system of the earlier method, where n is the number of subintervals in the partition. Further, there is a requirement of less regularity on exact solution in this method. The results are validated with numerical examples. Finally, instability behavior of the solution is numerically captured with this method.