Linear Galerkin Finite Element Method Research Articles

A fast linearized Galerkin finite element method for the nonlinear multi-term time fractional wave equation

This paper is concerned with a class of nonlinear multi-term time fractional wave equations. Based on the Galerkin finite element method (FEM) in space and the L2-1σ formula in time, combined with the corresponding fast algorithm, an efficient fully discrete scheme is constructed. The unconditional optimal convergence property in H1-norm of the proposed scheme is discussed by utilizing the time-space splitting technique. Furthermore, the optimal L2-norm convergence and the H1-norm superconvergence results of the method are obtained. Finally, several illustrative numerical experiments are performed to verify the theoretical results. Compared with the existing works, studying the convergence of FEMs for the nonlinear fractional wave equation is a great challenging task, and there are significant differences in research approaches.

Unconditionally Superconvergence Error Analysis of an Energy-Stable and Linearized Galerkin Finite Element Method for Nonlinear Wave Equations

Unconditionally Superconvergence Error Analysis of an Energy-Stable and Linearized Galerkin Finite Element Method for Nonlinear Wave Equations

Linearized transformed [formula omitted] finite element methods for semi-linear time-fractional parabolic problems

A linearized Galerkin finite element method is presented for numerically solving the semi-linear time-fractional parabolic problems, whose solutions always display a initial weak singularity. The transformed L1 scheme based on a change of variable is used to approximate Caputo derivatives and the finite element approximations to the spatial variables. By the temporal-spatial error splitting argument, unconditionally optimal error estimates of the proposed schemes are proved. Finally, several numerical experiments are given to demonstrate our theoretical results.

Unconditionally superconvergent error estimates of a linearized Galerkin finite element method for the nonlinear thermistor problem

Unconditionally superconvergent error estimates of a linearized Galerkin finite element method for the nonlinear thermistor problem

Unconditionally optimal convergence of a linearized Galerkin FEM for the nonlinear time-fractional mobile/immobile transport equation

In this paper, a linearized Galerkin finite element method (FEM) is discussed for solving the nonlinear time-fractional mobile/immobile transport equation. Utilizing the temporal–spatial error splitting argument, we derive the optimal L2-norm error estimate without the stepsize restriction condition τ=O(hd4). The key point in our analysis is to obtain the unconditionally optimal error estimate between the solutions of the time-discrete system and continuous problem in H2-norm, with which, we prove the boundedness of the fully discrete finite element solution in L∞-norm by using induction method. Then, the unconditionally optimal error estimate in L2-norm can be obtained in the usual way. Finally, three numerical examples in both two and three dimensional spaces are given to illustrate the correctness of our theoretical analysis.

Superconvergence analysis of nonconforming EQ1rot element for nonlinear parabolic integro‐differential equation

In this paper, superconvergent analysis for nonlinear parabolic integro‐differential equation is studied with nonconforming element. A linearized Galerkin finite element method is introduced, and a time‐discrete system is constructed to split the error into two parts. On the one hand, a rigorous analysis for the regularity of the time‐discrete system is presented based on the temporal error estimation skillfully. On the other hand, the spatial error is deduced τ‐independently with the above achievements. With the help of typical characters of this element, the superclose result of order in the broken H1‐norm is derived without any restriction of τ. At last, numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Error estimates for a robust finite element method of two-term time-fractional diffusion-wave equation with nonsmooth data

In this paper, we consider a two-term time-fractional diffusion-wave equation which involves the fractional ordersα∈ (1, 2) andβ∈ (0, 1), respectively. By using piecewise linear Galerkin finite element method in space and convolution quadrature based on second-order backward difference method in time, we obtain a robust fully discrete scheme. Error estimates for semidiscrete and fully discrete schemes are established with respect to nonsmooth data. Numerical experiments for two-dimensional problems are provided to illustrate the efficiency of the method and conform the theoretical results.

Two efficient Galerkin finite element methods for the modified anomalous subdiffusion equation

ABSTRACT In this paper, we consider the numerical approximation of the modified anomalous subdiffusion model which involves the Riemann–Liouville derivatives in time. We propose two robust fully discrete finite element methods by employing the piecewise linear Galerkin finite element method in space and the convolution quadrature in time generated by the backward Euler and the second-order backward difference methods. The error estimates for semidiscrete and fully discrete schemes are investigated with respect to the data regularity. Furthermore, we numerically compare our numerical schemes with a Crank–Nicolson finite element method to illustrate the efficiency of our methods and confirm the theoretical results.

A family of new globally convergent linearization schemes for solving Richards' equation

The Richards' equation models the water flow through porous media in groundwater aquifers and petroleum reservoir simulations. This is a degenerate parabolic differential equation, in which no analytical solution is known. Most numerical methods use implicit time schemes supporting large time steps, but involving near degenerate nonlinear systems. To solve them, we need robust and efficient linearization schemes. Apart from Newton's method, which fails to converge for large time steps and small mesh sizes, recently the globally convergent first order L-scheme was developed, which approximates the derivative by a global upper bound improving Picard's scheme.In this paper, we extend this method to build a family of robust efficient globally convergent first-order linearization schemes by using a sequence of real functions Ln, n≥0. We solve the space variable by a piecewise linear Galerkin finite element method. The time discretization is based on the Forward-Backward Euler method in the term of hydraulic conductivity K, whereas the remaining terms of the equation dealt with the Backward Euler approximation. We get five schemes improving the convergence of L-schemes and are getting closer to a second-order convergent method. We prove the time iteration convergence in the H1(Ω) norm, and we test the schemes on numerical benchmarks to compare them with the L-scheme and Newton's scheme. Among the new schemes, four of them are globally convergent and one scheme is a hybrid case with Newton's scheme. Our results show that among the new schemes, Modified Generalized Linear Scheme (MGLS) obtains the best convergence rates (up to four times that of the L-scheme) using fewer iteration steps and less overall computing time. Finally, our hybrid scheme is robust since it converges when Newton's scheme does not and uses similar computing time, which makes it a good alternative for locally convergent second-order schemes. More methods can be developed using this framework with other Ln sequences and spatial discretizations.

Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow

Particle tracking is a computationally advantageous and fast scheme to determine travel times and trajectories in subsurface hydrology. Accurate particle tracking requires element-wise mass-conservative, conforming velocity fields. This condition is not fulfilled by the standard linear Galerkin finite element method (FEM). We present a projection, which maps a non-conforming, element-wise given velocity field, computed on triangles and tetrahedra, onto a conforming velocity field in lowest-order Raviart-Thomas-Nédélec (mathcal {RTN}_{0}) space, which meets the requirements of accurate particle tracking. The projection is based on minimizing the difference in the hydraulic gradients at the element centroids between the standard FEM solution and the hydraulic gradients consistent with the mathcal {RTN}_{0} velocity field imposing element-wise mass conservation. Using the conforming velocity field in mathcal {RTN}_{0} space on triangles and tetrahedra, we present semi-analytical particle tracking methods for divergent and non-divergent flow. We compare the results with those obtained by a cell-centered finite volume method defined for the same elements, and a test case considering hydraulic anisotropy to an analytical solution. The velocity fields and associated particle trajectories based on the projection of the standard FEM solution are comparable to those resulting from the finite volume method, but the projected fields are smoother within zones of piecewise uniform hydraulic conductivity. While the mathcal {RTN}_{0}-projected standard FEM solution is thus more accurate, the computational costs of the cell-centered finite volume approach are considerably smaller.

Incomplete iterative solution of subdiffusion

In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order $$\alpha \in (0,1)$$ in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time level chosen properly, the error estimates are nearly identical with that for the exact linear solver, and the theoretical findings provide guidelines on the choice. Illustrative numerical results are presented to complement the theoretical analysis.

Linearized Galerkin FEMs for Nonlinear Time Fractional Parabolic Problems with Non-smooth Solutions in Time Direction

A Newton linearized Galerkin finite element method is proposed to solve nonlinear time fractional parabolic problems with non-smooth solutions in time direction. Iterative processes or corrected schemes become dispensable by the use of the Newton linearized method and graded meshes in the temporal direction. The optimal error estimate in the $$L^2$$ -norm is obtained without any time step restrictions dependent on the spatial mesh size. Such unconditional convergence results are proved by including the initial time singularity into concern, while previous unconditional convergent results always require continuity and boundedness of the temporal derivative of the exact solution. Numerical experiments are conducted to confirm the theoretical results.

Galerkin FEM for a time-fractional Oldroyd-B fluid problem

We consider the numerical approximation of a generalized fractional Oldroyd-B fluid problem involving two Riemann-Liouville fractional derivatives in time. We establish regularity results for the exact solution which play an important role in the error analysis. A semidiscrete scheme based on the piecewise linear Galerkin finite element method in space is analyzed, and optimal with respect to the data regularity error estimates are established. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are investigated and related error estimates for smooth and nonsmooth data are derived. Numerical experiments are performed with different values of the problem parameters to illustrate the efficiency of the method and confirm the theoretical results.

Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

Abstract In this work we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation that involves a fractional derivative of order $\alpha \in (0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational-type discretization. With a space mesh size $h$ and time stepsize $\tau $ we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(\tau ^{\min ({1}/{2}+\alpha -\epsilon ,1)}+h^2)$ in the discrete $L^2(0,T;L^2(\varOmega ))$ norm and $O(\tau ^{\alpha -\epsilon }+\ell _h^2h^2)$ in the discrete $L^{\infty }(0,T;L^2(\varOmega ))$ norm, with any small $\epsilon>0$ and $\ell _h=\ln (2+1/h)$. The analysis relies essentially on the maximal $L^p$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.

Unconditional superconvergence analysis of a linearized Galerkin FEM for nonlinear hyperbolic equations

Unconditional superconvergence analysis for nonlinear hyperbolic equations with bilinear finite element is studied. A linearized Galerkin finite element method (FEM) is developed and a time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, a rigorous analysis for the regularity of the time-discrete system is presented based on the temporal error estimation skillfully. On the other hand, the numerical solution ‖Uhn‖0,∞+‖∂̃tUhn‖0,∞ is bounded by the spatial error which is deduced with the help of the Ritz projection operator. The superclose and global superconvergence estimates of u with order O(h2+τ2) in H1-norm are derived without any restriction of τ through the relationship between the interpolation operator and the Ritz projection operator. At last, numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ, the time step.

Unconditional superconvergence analysis for nonlinear hyperbolic equation with nonconforming finite element

Nonlinear hyperbolic equation is studied by developing a linearized Galerkin finite element method (FEM) with nonconforming EQ1rot element. A time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully which leads to the analysis for the regularity of the time-discrete system. The spatial error is derived τ-independently with order O(h2+hτ) in broken H1-norm. The final unconditional superclose result of u with order O(h2+τ2) is deduced based on the above achievements. The two typical characters of this nonconforming EQ1rot element (see Lemma 1 below) play an important role in the procedure of proof. At last, a numerical example is provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ, the time step.

A New Mixed Formulation and Efficient Numerical Solution of Ginzburg--Landau Equations Under the Temporal Gauge

In this paper, we present a new numerical approach to the time-dependent Ginzburg--Landau (GL) equations under the temporal gauge (zero electric potential gauge). The approach is based on a mixed formulation of the GL equations, which consists of two parabolic equations for the order parameter $\psi$ and the magnetic field $\sigma = \mathrm{curl} \, \mathbf{A}$, respectively, and a vector ordinary differential equation for the magnetic potential $\mathbf{A}$. A fully linearized Galerkin finite element method is presented for solving the mixed GL system. The new approach offers many advantages on both accuracy and efficiency over existing methods. In particular, the equations for $\psi$ and $\sigma$ are uniformly parabolic and, therefore, the method provides optimal-order accuracy for the two physical components $\psi$ and $\sigma$. Since in the temporal direction, a fully linearized backward Euler scheme is used for $\psi$ and $\sigma$ and a forward Euler scheme is used for $\mathbf{A}$, respectively, the system is fully decoupled and at each time step, the three variables $\psi$, $\sigma$, and $\mathbf{A}$ can be solved simultaneously. Moreover, we present numerical comparisons with two commonly used Galerkin methods for the GL equations under the temporal gauge and the Lorentz gauge, respectively. Our numerical results show that the new approach requires fewer iterations for solving the linear systems arising at each time step and the computational cost for the vector ODE seems neglectable. Several numerical examples in both two- and three-dimensional spaces are investigated.

Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.

A modified Galerkin FEM for 1D Helmholtz equations

A method is presented that aims to eliminate the numerical errors inherent in the standard Galerkin finite element method (GFEM) for solving homogeneous Helmholtz equations. An error analysis of the standard GFEM with linear elements is first performed by using the concept of truncation error in finite difference methods, and then the truncation error expression is obtained. A linear GFEM with an artificial stiffness is proposed to solve the Helmholtz equation after investigating the effect of the error on numerical solution. The proposed method is essentially as straightforward as the standard GFEM and thus requires almost no additional computational effort. Numerical results show that present pollution error decreases by 90% compared with that of the standard GFEM.