Stabilized discontinuous finite element approximations for Stokes equations

In this paper, we derive two stabilized discontinuous finite element formulations, symmetric and nonsymmetric, for the Stokes equations and the equations of the linear elasticity for almost incompressible materials. These methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity/displacements are imposed in a weak sense via Lagrange multipliers. For both methods, almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure. Optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in L²$L^2$ norm are obtained, as well as one in a mesh-dependent norm for the velocity in both symmetric and nonsymmetric formulations.     

Finite element approximation to nonlinear coupled thermal problem

A nonlinear coupled elliptic system modelling a large class of engineering problems was discussed in [A.F.D. Loula, J. Zhu, Finite element analysis of a coupled nonlinear system, Comp. Appl. Math. 20 (3) (2001) 321–339; J. Zhu, A.F.D. Loula, Mixed finite element analysis of a thermally nonlinear coupled problem, Numer. Methods Partial Differential Equations 22 (1) (2006) 180–196]. The convergence analysis of iterative finite element approximation to the solution was done under an assumption of ‘small’ solution or source data which guarantees the uniqueness of the nonlinear coupled system. Generally, a nonlinear system may have multiple solutions. In this work, the regularity of the weak solutions is further studied. The nonlinear finite element approximations to the nonsingular solutions are then proposed and analyzed. Finally, the optimal order error estimates in H¹$H^1$-norm and L²$L^2$-norm as well as in W^1,p$W^{1,p}$-norm and L^p$L^p$-norm are obtained.     

Second-order implicit–explicit scheme for the Gray–Scott model

A second-order scheme for the Gray–Scott (GS) model used to describe the pattern formation is studied. The linear part of the GS equation for the time derivative and the viscous terms is discretized implicitly, while the other (or nonlinear) part of the GS equation explicitly. Galerkin finite element approximation methods are presented and analyzed, as well as methods for solving the resulting system of equations. The optimal L²$L^2$-norm error estimates are derived. Numerical experiments are presented.     

Nonconforming finite element method for stochastic Stokes equations

In this paper, we study nonconforming finite element method for stochastic Stokes equation driven by white noise. We apply “green function framework” and standard duality technique to study the error estimate for velocity in L²$L^2$-norm and for pressure in H^-1$H^{-1}$-norm. Finally, numerical experiment proves our theoretical results.     

Analysis of a splitting method for incompressible inviscid rotational flow problems

This paper is devoted to analyze a splitting method for solving incompressible inviscid rotational flows. The problem is first recast into the velocity–vorticity–pressure formulation by introducing the additional vorticity variable, and then split into three consecutive subsystems. For each subsystem, the L²$L^2$ least-squares finite element approach is applied to attain accurate numerical solutions. We show that for each time step this splitting least-squares approach exhibits an optimal rate of convergence in the H¹$H^1$ norm for velocity and pressure, and a suboptimal rate in the L²$L^2$ norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates.     

Two-grid methods for finite volume element approximations of nonlinear parabolic equations

Two-grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method. The methods are based on one coarse-grid space and one fine-grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine-grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H³|lnH|)$h=O\left(H^3|lnH|\right)$. As a result, solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation.     

Superconvergence of mixed covolume method for elliptic problems on triangular grids

In this paper, we consider the superconvergence of a mixed covolume method on the quasi-uniform triangular grids for the variable coefficient-matrix Poisson equations. The superconvergence estimates between the solution of the mixed covolume method and that of the mixed finite element method have been obtained. With these superconvergence estimates, we establish the superconvergence estimates and the L^∞$L^{\infty }$-error estimates for the mixed covolume method for the elliptic problems. Based on the superconvergence of the mixed covolume method, under the condition that the triangulation is uniform, we construct a post-processing method for the approximate velocity which improves the order of approximation of the approximate velocity.     

Error estimation of a quadratic finite volume method on right quadrangular prism grids

In this paper, we develop a finite volume element method with affine quadratic bases on right quadrangular prism meshes for three-dimensional elliptic boundary value problems. The optimal H¹$H^1$-norm error estimate of second order accuracy is proved under certain assumptions about the meshes. Numerical results are presented to illustrate the theoretical analysis.     

A remark on least-squares mixed element methods for reaction–diffusion problems

In this paper, we propose a least-squares mixed element procedure for a reaction–diffusion problem based on the first-order system. By selecting the least-squares functional properly, the resulting procedure can be split into two independent symmetric positive definite schemes, one of which is for the unknown variable and the other of which is for the unknown flux variable, which lead to the optimal order H¹(Ω)$H^1(\Omega )$ and L²(Ω)$L^2(\Omega )$ norm error estimates for the primal unknown and optimal H(div;Ω)$H(\mathrm{div};\Omega )$ norm error estimate for the unknown flux. Finally, we give some numerical examples.     

Linear and bilinear immersed finite elements for planar elasticity interface problems

This article is to discuss the linear (which was proposed in  and ) and bilinear immersed finite element (IFE) methods for solving planar elasticity interface problems with structured Cartesian meshes. Basic features of linear and bilinear IFE functions, including the unisolvent property, will be discussed. While both methods have comparable accuracy, the bilinear IFE method requires less time for assembling its algebraic system. Our analysis further indicates that the bilinear IFE functions are guaranteed to be applicable to a larger class of elasticity interface problems than linear IFE functions. Numerical examples are provided to demonstrate that both linear and bilinear IFE spaces have the optimal approximation capability, and that numerical solutions produced by a Galerkin method with these IFE functions for elasticity interface problem also converge optimally in both L²$L^2$ and semi-H¹$H^1$ norms.     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.     

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

This paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem by mixed finite elements. By means of integral expansion technique, the asymptotic error expansions for the approximations of the Stokes eigenvalue problem by Bernadi–Raugel element and _Q2-_P1$Q_2-P_1$ element are given. Based on such expansions, the extrapolation technique is applied to improve the accuracy of the approximations.     

A discontinuous Galerkin method on refined meshes for the two-dimensional time-harmonic Maxwell equations in composite materials

In this paper, a discontinuous Galerkin method for the two-dimensional time-harmonic Maxwell equations in composite materials is presented. The divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized. Due to an appropriate mesh refinement near exterior and interior corners, the singular behaviour of the electromagnetic field is taken into account. Optimal error estimates in a discrete energy norm and in the L²$L^2$-norm are proved in the case where the exact solution is singular.     

Convergence analysis for least-squares finite element approximations of second-order two-point boundary value problems

In this paper, a C⁰$C^0$ least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results.     

The finite volume method based on stabilized finite element for the stationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two-dimensional stationary Navier–Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation for the problem. We obtain the well-posedness of the FVM based on stabilized finite element for the stationary Navier–Stokes equations. Moreover, for quadrilateral and triangular partition, the optimal H¹$H^1$ error estimate of the finite volume solution _uh$u_h$ and L²$L^2$ error estimate for _ph$p_h$ are introduced. Finally, we provide a numerical example to confirm the efficiency of the FVM.     

A two‐grid method for characteristic expanded mixed finite element solution of miscible displacement problem

A combined method consisting of mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics(CEMFEM) for the concentration equation is presented to solve the coupled system of incompressible miscible displacement problem. To solve the resulting nonlinear system of equations efficiently, the two‐grid algorithm relegates all of the Newton‐like iterations to grids much coarser than the original one, with no loss in order of accuracy. It is shown that coarse space can be extremely coarse and our algorithm achieve asymptotically optimal approximation when the mesh sizes satisfy $H=O(h^{\frac{1}{4}})$. Numerical experiment is provided to confirm our theoretical results.     

Time discretization and quantization methods for optimal multiple switching problem

In this paper, we study probabilistic numerical methods based on optimal quantization algorithms for computing the solution to optimal multiple switching problems with regime-dependent state process. We first consider a discrete-time approximation of the optimal switching problem, and analyse its rate of convergence. Given a time step h$h$, the error is in general of order (hlog(1/h))^1/2${(hlog(1/h))}^{1/2}$, and of order h^1/2$h^{1/2}$ when the switching costs do not depend on the state process. We next propose quantization numerical schemes for the space discretization of the discrete-time Euler state process. A Markovian quantization approach relying on the optimal quantization of the normal distribution arising in the Euler scheme is analysed. In the particular case of uncontrolled state process, we describe an alternative marginal quantization method, which extends the recursive algorithm for optimal stopping problems as in Bally (2003) [1]. A priori L^p$L^p$-error estimates are stated in terms of quantization errors. Finally, some numerical tests are performed for an optimal switching problem with two regimes.