H1-Galerkin Mixed Finite Element Research Articles

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.

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

An [formula omitted]-Galerkin mixed finite element method for identification of time dependent parameters in parabolic problems

A direct method of identification of time dependent parameters in a linear parabolic boundary value problem with over-specified total internal energy involves the flux at the boundary, and an H1 mixed formulation seems to be more suitable than the standard methods for such class of nonlocal problems. Therefore, this paper develops and analyses an H1-Galerkin mixed finite element method. Optimal error estimates in both primary and flux variables are derived in semidiscrete case. Moreover, a priori error estimate for the parameters is established. Based on linearised backward Euler method, a completely discrete scheme is proposed and optimal error analysis is derived. The results of the numerical experiments show the efficacy of the proposed method and confirm our theoretical results.

Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model

In this article, a time two-mesh (TT-M) algorithm combined with the H1-Galerkin mixed finite element (FE) method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model, which is faster than the H1-Galerkin mixed FE method. The Crank-Nicolson scheme with TT-M algorithm is used to discretize the temporal direction at time $t_{n+\frac {1}{2}}$ , the FBN-𝜃 formula is developed to approximate the distributed-order derivative, and the H1-Galerkin mixed FE method is used to approximate the spatial direction. TT-M mixed element algorithm mainly covers three steps: first, the mixed finite element solution of the nonlinear coupled system on the time coarse mesh ΔtC is calculated; next, based on the numerical solution obtained in the first step, the numerical solution of the nonlinear coupled system on time fine mesh ΔtF is obtained by using Lagrange’s interpolation formula; finally, the numerical solution of the linearized system on time fine mesh ΔtF is solved by using the results in the second step. The existence and uniqueness of the solution for our numerical scheme are shown. Moreover, the stability and a priori error estimate are analyzed in detail. Furthermore, numerical examples with smooth and nonsmooth solutions are given to validate our method.

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 priori and a posteriori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations

In this paper, we investigate a priori and a posteriori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Based on two new elliptic projections, we derive a priori error estimates both for the control variable, the state variable and the co-state variable. The related a priori error estimates for the new projections error are also established. Moreover, a posteriori error estimates for all variables are derived via energy method. Such a posteriori error estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

Two temporal second-order H1-Galerkin mixed finite element schemes for distributed-order fractional sub-diffusion equations

In this paper, a special point is found for the interpolation approximation of the distributed order fractional derivatives to achieve at least second-order accuracy. Then, two H1-Galerkin mixed finite element schemes combined with the higher accurate interpolation approximation are introduced and analyzed to solve the distributed order fractional sub-diffusion equations. The stable results, which just depend on initial value and source item, are derived. Some a priori estimates with optimal order of convergence both for the unknown function and its flux are established rigorously. It is shown that the H1-Galerkin mixed finite element approximations have the same rates of the convergence as in the classical mixed finite element method, but without LBB consistency condition and quasiuniformity requirement on the finite element mesh. Finally, some numerical experiments are presented to show the efficiency and accuracy of H1-Galerkin mixed finite element schemes.

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.

New elliptic projections and a priori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations

In this paper, we discuss a priori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Both semidiscrete and fully discrete schemes are considered. Based on some new elliptic projections, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. The related a priori error estimates for the new projections error are also established. A numerical example is given to demonstrate the theoretical results.

Second-order approximation scheme combined with [formula omitted]-Galerkin MFE method for nonlinear time fractional convection–diffusion equation

In this article, a second-order approximation scheme combined with an H1-Galerkin mixed finite element (MFE) method for solving nonlinear convection–diffusion equation with time fractional derivative is proposed and analyzed. By introducing an auxiliary variable, a coupled system is formulated, then the spatial direction is approximated by H1-Galerkin MFE method and the temporal fractional derivative and integer derivative are discretized by second-order weighted and shifted Grünwald difference (WSGD) formula and linearized second-order difference scheme, respectively. The optimal priori error estimates in L2 and H1-norm for the unknown function and the auxiliary variable with second-order convergent rate in time are obtained. Compared to the commonly used L1-approximation with (2−α)th-order convergence rate, our method can arrive at the order 2 in time. What is more, compared with the standard finite element method, our method can well approximate the auxiliary variable. Finally, the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis.

Superconvergence analysis of an [formula omitted]-Galerkin mixed finite element method for Sobolev equations

An H1-Galerkin mixed finite element method (MFEM) is discussed for the Sobolev equations with the bilinear element and zero order Raviart–Thomas element (Q11+Q10×Q01). The existence and uniqueness of the solutions about the approximation scheme are proved. Two new important lemmas are given by using the properties of the integral identity and the Bramble–Hilbert lemma, which lead to the superclose results of order O(h2) for original variable u in H1 norm and flux q→ in H(div;Ω) norm under semi-discrete scheme. Furthermore, two new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained. On the other hand, a second order fully-discrete scheme with superclose property O(h2+τ2) is also proposed. At last, numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.

FINITE DIFFERENCE/<i>H</i><sup>1</sup>-GALERKIN MFE PROCEDURE FOR A FRACTIONAL WATER WAVE MODEL

In this article, an H1-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and L1 formula are applied to approximate integer derivative and Caputo fractional derivative with order 1/2, respectively, and H1-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both H1-norm and L2-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method.

A priori and a posteriori error estimates of [formula omitted]-Galerkin mixed finite element methods for elliptic optimal control problems

In this paper, we investigate numerical approximations of H1-Galerkin mixed finite element methods for elliptic optimal control problems. The presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables. Moreover, the matching relation (i.e., LBB-condition) between the mixed element spaces Vh and Wh is not necessary, thus, we can choose the approximation spaces more flexibly. The state and co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the standard finite element spaces, the control variable is approximated by piecewise constant functions. We derive a priori and a posteriori error estimates for the control variable, the state variables and the adjoint state variables. Finally, some numerical examples are given to demonstrate the theoretical results about a priori error estimates.

Convergence of H 1-Galerkin mixed finite element method for parabolic problems with reduced regularity on initial data

We study the convergence of H1-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming less regularity on initial data. More precisely, for the spatially discrete scheme, error estimates of order \(\mathcal{O}\)(h2t−1/2) for positive time are established assuming the initial function p0 ∈ H2(Ω) ∩ H01 (Ω). Further, we use energy technique together with parabolic duality argument to derive error estimates of order \(\mathcal{O}\)(h2t−1) when p0 is only in H01 (Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.

The Numerical Analysis and Simulation of a Linearized Crank-Nicolson <i>H</i><sup>1</sup>-GMFEM for Nonlinear Coupled BBM Equations

In this article, the numerical scheme of a linearized Crank-Nicolson (C-N) method based on H1-Galerkin mixed finite element method (H1-GMFEM) is studied and analyzed for nonlinear coupled BBM equations. In this method, the spatial direction is approximated by an H1-GMFEM and the time direction is discretized by a linearized Crank-Nicolson method. Some optimal a priori error results are derived for four important variables. For conforming the theoretical analysis, a numerical test is presented.

Domain decomposition procedures combined with [formula omitted]-Galerkin mixed finite element method for parabolic equation

Non-overlapping domain decomposition procedures are considered for parabolic equation. These procedures are combined with using H1-Galerkin mixed finite element method in the sub-domains to approximate the primary variable u and its flux σ simultaneously. Explicit calculations are built by using integral mean methods to present the inter-domain boundary conditions for the flux. Thus, the parallelism can be achieved by these procedures. Two approximation schemes are established. Time step constraints are proved necessary to preserve stability, which are less severe than that of fully explicit Galerkin finite element method. The mixed finite element spaces are allowed to be of different polynomial degrees and not subject to the LBB-consistency condition. New nonstandard elliptic projections are defined and analyzed. Optimal error estimates for the variable u in H1-norm and its flux σ in L2-norm and are derived for these schemes. Numerical experiments are presented to confirm the theoretical results.

Superconvergence of an [formula omitted]-Galerkin nonconforming mixed finite element method for a parabolic equation

By choosing a suitable pair of approximating spaces, an H1-Galerkin nonconforming mixed finite element method (FEM) is proposed for a class of parabolic equations under semi-discrete, backward Euler and Crank–Nicolson fully-discrete schemes, in which the famous EQ1rot element and zero order Raviart–Thomas element are used to approximate the primitive solution u and the flux p→=∇u, respectively. Based on special characters of the elements considered, the corresponding optimal order error estimates for u in broken H1-norm and p→ in H(div)-norm are obtained for the above schemes. Furthermore, the global superconvergence results are derived through the postprocessing technique. The numerical results show the validity of the theoretical analysis.

Convergence analysis for H1-Galerkin mixed finite element approximation of one nonlinear integro-differential model

In this paper we investigate H1-Galerkin mixed finite element approximation of one nonlinear integro-differential equation. This method possesses some advantages such as approximating the unknown function and its gradient simultaneously as well as the finite element spaces being free of LBB condition. A priori error estimates of the unknown function and its gradient are derived for both semi-discrete and fully discrete schemes. A numerical example is presented to illustrate the theoretical findings.