Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation

In this paper, a class of nonlinear Riesz space-fractional Schrodinger equations are considered. Based on the standard Galerkin finite element method in space and a relaxation-type difference method in time, a fully discrete system is constructed. This scheme avoids solving the nonlinear systems and preserves the mass and energy very well. By the Brouwer fixed-pointed theorem, the unique solvability of the discrete system is proved. Moreover, we focus on a rigorous analysis of the optimal convergence properties for the fully discrete system. Finally, some numerical examples are given to validate the theoretical analysis.

Strong convergence rates of Galerkin finite element methods for SWEs with cubic polynomial nonlinearity

In this paper, a class of nonlinear Riesz space-fractional Schrodinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind, with a weakly singular kernel in the time variable, and which also involves partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used at each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, which can be used as the basis for an adaptive strategy.

ADAPTIVE DISCRETIZATION OF AN INTEGRO-DIFFERENTIAL EQUATION MODELING QUASI-STATIC FRACTIONAL ORDER VISCOELASTICITY

A comprehensive groundwater solute transport simulator is developed based on the modified method of characteristics (MMOC) combined with the Galerkin finite element method for the transport equation and the mixed finite element (MFE) method for the groundwater flow equation. The preconditioned conjugate gradient algorithm is used to solve the two large sparse algebraic system of equations arising from the MMOC and MFE discretizations. The MMOC takes time steps in the direction of flow, along the characteristics of the velocity field of the total fluid. The physical diffusion and dispersion terms are treated by a standard finite element scheme. The crucial aspect of the MMOC technique is that it looks backward in time, along an approximate flow path, instead of forward in time as in many method of characteristics or moving mesh techniques. The MFE procedure involves solving for both the hydraulic head and the specific discharge simultaneously. One order of convergence is gained by the MFE method, as compared with other standard finite element methods, and therefore more accurate velocity fields are simulated. The overall advantages of the MMOC‐MFE method include minimum numerical oscillation or grid orientation problems under steep concentration gradient simulations, and material balance errors are greatly reduced due to a very accurate velocity simulation by the MFE method. In addition, much larger time steps with Courant number well in excess of 1, as compared with the standard Galerkin finite element method, can be taken on a fixed spatial grid system without significant loss of accuracy.

In this article, the coupled Schrödinger–Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M Crank–Nicolson scheme, and then the numerical scheme of TT-M finite element (FE) system is formulated. The method includes three main steps: for the first step, the nonlinear system is solved on the coarse time mesh; for the second step, by an interpolation formula, the numerical solutions at the fine time mesh point are computed based on the numerical solutions on the coarse mesh system; for the last step, the linearized temporal fine mesh system is constructed based on Taylor’s formula for two variables, and then the TT-M FE solutions can be obtained. Furthermore, theory analyses on the TT-M system including the stability and error estimations are conducted. Finally, a large number of numerical examples are provided to verify the accuracy of the algorithm, the correctness of theoretical results, and the computational efficiency with a comparison to the numerical results calculated by using the standard FE method.

In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process { u ( t ) } t ∈ [ 0 , T ] \{u(t)\}_{t\in [0,T]} satisfying a stochastic linear evolution equation with a positive-type memory term driven by an additive Gaussian noise. The equation can be written in an abstract form as \[ d u + ( ∫ 0 t b ( t − s ) A u ( s ) d s ) d t = d W Q , t ∈ ( 0 , T ] ; u ( 0 ) = u 0 ∈ H , \mathrm {d} u + \left ( \int _0^t b(t-s) Au(s) \, \mathrm {d} s \right )\, \mathrm {d} t = \mathrm {d} W^{_Q},~t\in (0,T]; \quad u(0)=u_0 \in H, \] where W Q W^{_Q} is a Q Q -Wiener process on H = L 2 ( D ) H=L^2({\mathcal D}) and where the main example of b b we consider is given by \[ b ( t ) = t β − 1 / Γ ( β ) , 0 > β > 1. b(t) = t^{\beta -1}/\Gamma (\beta ), \quad 0 > \beta >1. \] We let A A be an unbounded linear self-adjoint positive operator on H H and we further assume that there exist α > 0 \alpha >0 such that A − α A^{-\alpha } has finite trace and that Q Q is bounded from H H into D ( A κ ) D(A^\kappa ) for some real κ \kappa with α − 1 β + 1 > κ ≤ α \alpha -\frac {1}{\beta +1}>\kappa \leq \alpha . The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter Δ t = T / n \Delta t =T/n ), and a standard continuous finite element method in space (parameter h h ). Let u n , h u_{n,h} be the discrete solution at T = n Δ t T=n\Delta t . We show that ( E ‖ u n , h − u ( T ) ‖ 2 ) 1 / 2 = O ( h ν + Δ t γ ) , \begin{equation*} \left ( \mathbb {E} \| u_{n,h} - u(T)\|^2 \right )^{1/2}={\mathcal O}(h^{\nu } + \Delta t^\gamma ), \end{equation*} for any γ > ( 1 − ( β + 1 ) ( α − κ ) ) / 2 \gamma > (1 - (\beta +1)(\alpha - \kappa ))/2 and ν ≤ 1 β + 1 − α + κ \nu \leq \frac {1}{\beta +1}-\alpha +\kappa .

SummaryWe demonstrate the potential of collocation methods for efficient higher‐order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra‐weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher‐order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher‐order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical solution of two-carrier hydrodynamic semiconductor device equations employing a stabilized finite element method

The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order O ( ε 1 / 4 N − 1 + N − 1 ln ⁡ N ) . Numerical tests are carried out in support of theoretical findings.

In contrast with the diffusion equation which smoothens the initial data to C ∞ C^\infty for t > 0 t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.

In this paper, we consider an initial boundary value problem for nonlocal-in-time parabolic equations involving a nonlocal in time derivative. We first show the uniqueness and existence of the weak solution of the nonlocal-in-time parabolic equation, and also the spatial smoothing properties. Moreover, we develop a new framework to study the local limit of the nonlocal model as the horizon parameter δ approaches 0. Exploiting the spatial smoothing properties, we develop a semi-discrete scheme using standard Galerkin finite element method for the spatial discretization, and derive error estimates dependent on data smoothness. Finally, extensive numerical results are presented to illustrate our theoretical findings.

We present a fully discrete hp -version numerical method for second-order linear parabolic equations, combining a continuous Petrov–Galerkin (CPG) time-stepping scheme with a standard continuous Galerkin (CG) finite element method in space. Our analysis provides a priori error estimates in the L 2 ( L 2 )- and L ∞ ( L 2 )-norms, where all constants are fully robust – independent of temporal and spatial discretization parameters. For solutions with initial singularities at t = 0, exponential temporal convergence is achieved through geometrically graded time meshes and linearly increasing polynomial degrees. Numerical experiments confirm the theoretical convergence rates.