Collocation method with Lagrange polynomials for variable-order time-fractional advection–diffusion problems

In this study, we proposed a numerical technique to solve a class of variable-order time-fractional advection–diffusion equations (VOTFADEs) by applying an operational matrix of differentiation based on fractional-order Lagrange polynomials (FOLPs). The variable-order fractional derivative is assumed to be Caputo's derivative. Using the operational matrix and collocation method, the advection–diffusion equation can be reduced to an algebraic system of equations that can be solved using Newton's iterative method. Error analysis also has been carried out for the proposed method. The current approach is simple to use and computer oriented and provides highly accurate approximate solutions. The effectiveness and accuracy of the proposed method are demonstrated using a few numerical examples.     

A P0–P0 weak Galerkin finite element method for solving singularly perturbed reaction–diffusion problems

This paper investigates the lowest-order weak Galerkin finite element (WGFE) method for solving reaction–diffusion equations with singular perturbations in two and three space dimensions. The system of linear equations for the new scheme is positive definite, and one might readily get the well-posedness of the system. Our numerical experiments confirmed our error analysis that our WGFE method of the lowest order could deliver numerical approximations of the order O(h^1/2) and O(h) in H¹ and L² norms, respectively.     

A modified weak Galerkin finite element methods for convection–diffusion problems in 2D

In this paper, we develop a modified weak Galerkin finite element method on arbitrary grids for convection–diffusion problems in two dimensions based on our previous work (Wang et al., J Comput Appl Math 271, 319–327, 2014), in which we only considered second order Poisson equations and thus only introduced a modified weak gradient operator. This method, called MWG-FEM, is based on a modified weak gradient operator and weak divergence operator which is put forward in this paper. Optimal order error estimates are established for the corresponding MWG-FEM approximations in both a discrete \(H^1\) norm and the standard \(L^2\) norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the MWG-FEM.     

A parallel two-level finite element variational multiscale method for the Navier–Stokes equations

A combination method of two-grid discretization approach with a recent finite element variational multiscale algorithm for simulation of the incompressible Navier–Stokes equations is proposed and analyzed. The method consists of a global small-scale nonlinear Navier–Stokes problem on a coarse grid and local linearized residual problems in overlapped fine grid subdomains, where the numerical form of the Navier–Stokes equations on the coarse grid is stabilized by a stabilization term based on two local Gauss integrations at element level and defined by the difference between a consistent and an under-integrated matrix involving the gradient of velocity. By the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method.     

A novel finite volume method for the Riesz space distributed-order advection–diffusion equation

In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) equation. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional model. Next, the transformed multi-term fractional equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 < γ < 1 is transformed into a fractional integral form. An important contribution of our work is the use of nodal basis function to derive the discrete form of our model. The unique solvability of the scheme is also discussed and we prove that the Crank–Nicolson scheme is unconditionally stable and convergent with second-order accuracy. Finally, we give some examples to show the effectiveness of the numerical method.     

Mortar multiscale finite element methods for Stokes–Darcy flows

We investigate mortar multiscale numerical methods for coupled Stokes and Darcy flows with the Beavers–Joseph–Saffman interface condition. The domain is decomposed into a series of subdomains (coarse grid) of either Stokes or Darcy type. The subdomains are discretized by appropriate Stokes or Darcy finite elements. The solution is resolved locally (in each coarse element) on a fine scale, allowing for non-matching grids across subdomain interfaces. Coarse scale mortar finite elements are introduced on the interfaces to approximate the normal stress and impose weakly continuity of the normal velocity. Stability and a priori error estimates in terms of the fine subdomain scale $h$ and the coarse mortar scale $H$ are established for fairly general grid configurations, assuming that the mortar space satisfies a certain inf-sup condition. Several examples of such spaces in two and three dimensions are given. Numerical experiments are presented in confirmation of the theory.     

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”- and “DG”-norm formed by the “L ²(H ¹)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.     

Hybridized schemes of the discontinuous Galerkin method for stationary convection–diffusion problems

For stationary linear convection–diffusion problems, we construct and study a new hybridized scheme of the discontinuous Galerkin method on the basis of an extended mixed statement of the problem. Discrete schemes can be used for the solution of equations degenerating in the leading part and are stated via approximations to the solution of the problem, its gradient, the flow, and the restriction of the solution to the boundaries of elements. For the spaces of finite elements, we represent minimal conditions responsible for the solvability, stability, accuracy, and superconvergence of the schemes. A new procedure for the post-processing of solutions of HDG-schemes is suggested.     

Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger 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.     

A finite element method for two–parameter singularly perturbed problems in 2D

We consider a singularly perturbed elliptic problem with two small parameters posed on the unit square. Based on a decomposition of the solution, we prove uniform convergence of a finite element method in an energy norm. The method uses piecewise bilinear functions on a layer-adapted Shishkin mesh. Numerical results confirm our theoretical analysis. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori error estimates of finite element method for parabolic problems

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An Alternating-Direction Sinc–Galerkin method for elliptic problems

A new Alternating-Direction Sinc–Galerkin (ADSG) method is developed and contrasted with classical Sinc–Galerkin methods. It is derived from an iterative scheme for solving the Lyapunov equation that arises when a symmetric Sinc–Galerkin method is used to approximate the solution of elliptic partial differential equations. We include parameter choices (derived from numerical experiments) that simplify existing alternating-direction algorithms. We compare the new scheme to a standard method employing Gaussian elimination on a system produced using the Kronecker product and Kronecker sum, as well as to a more efficient algorithm employing matrix diagonalization. We note that the ADSG method easily outperforms Gaussian elimination on the Kronecker sum and, while competitive with matrix diagonalization, does not require the computation of eigenvalues and eigenvectors.     

A parameter robust Petrov–Galerkin scheme for advection–diffusion–reaction equations

In this paper, a singularly perturbed convection diffusion boundary value problem, with discontinuous diffusion coefficient is examined. In addition to the presence of boundary layers, strong and weak interior layers can also be present due to the discontinuities in the diffusion coefficient. A priori layer adapted piecewise uniform meshes are used to resolve any layers present in the solution. Using a Petrov–Galerkin finite element formulation, a fitted finite difference operator is shown to produce numerical approximations on this fitted mesh, which are uniformly second order (up to logarithmic terms) globally convergent in the pointwise maximum norm.