Error estimate of the P 1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations

We consider a finite element method for the penalty formulation of the time dependent Navier-Stokes equations. Usually the improper choice of the finite element space will lead that the error estimate (inversely) depends on the penalty parameter e{\epsilon}. We use the classical P ₁ nonconforming finite element space for the spatial discretization. Optimal e{\epsilon}-uniform error estimations for both velocity and pressure are obtained.     

A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations, where the time discretization is based on the Euler implicit/explicit scheme with some implicit linear terms and an explicit nonlinear term, and the finite element spatial discretization is based on the P₁b-P₁ element pair, which satisfies the discrete inf-sup condition. This method allows us to separate the computation of the velocity from the computation of the pressure with a larger time-step size Δt, so that the numerical velocity and the pressure are easily computed. An optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter ?, the time-step size Δt and the mesh size h satisfy the following stability conditions: ?c₁≤1, Δtκ₁≤1 and h²≤β₁Δt, respectively, for some positive constants c₁, κ₁ and β₁. Finally, some numerical tests to confirm the theoretical results of the penalty finite element method are provided.     

Numerical analysis of a modified finite element nonlinear Galerkin method

Summary. A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces X_H and X_h defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term. We analyze the stability and convergence rate of the method. Comparing with the standard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.Mathematics Subject Classification (2000): 35Q30, 65M60, 65N30, 76D05     

Penalty finite element approximations for the Stokes equations by L2 projection

This paper considers the penalty finite element method for the Stokes equations, based on some stable finite elements space pair (X_h, M_h) that do satisfy the discrete inf–sup condition. Theoretical results show that the penalty error converges as fast as one should expect from the order of the elements. Moreover, the penalty finite element method by L² projection can improve the penalty error estimates. Finally, we confirm these results by a series of numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd.     

Error estimates for two‐level penalty finite volume method for the stationary Navier–Stokes equations

Two‐level penalty finite volume method for the stationary Navier–Stokes equations based on the P₁ ? P₀ element is considered in this paper. The method involves solving one small penalty Navier–Stokes problem on a coarse mesh with mesh size H = ?^{1 / 4}h^{1 / 2}, a large penalty Stokes problem on a fine mesh with mesh size h, where 0 < ? < 1 is a penalty parameter. The method we study provides an approximate solution with the convergence rate of same order as the penalty finite volume solution (u_?h,p_?h), which involves solving one large penalty Navier–Stokes problem on a fine mesh with the same mesh size h. However, our method can save a large amount of computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

An error estimate for a finite volume scheme for a diffusion–convection problem on a triangular mesh

We study here a finite volume scheme for a diffusion-convection equation on an open bounded set Ω of ?², using a triangular mesh for the discretization of Ω. The 4-point numerical scheme is presented along with the geometrical assumptions on the mesh. An error estimate of order h on the discrete L² norm is obtained, where h denotes the “size” of the mesh. The proof uses an estimate of order h of the consistency error on the fluxes and an estimate of the number of edges of the mesh between one given triangle and the boundary Ω. © 1995 John Wiley & Sons, Inc.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Non-polynomials spline methods for solving a fourth-order parabolic equation

In this paper a fourth-order variable coefficient parabolic partial differential equation, that governs the behaviour of a vibrating beam, is solved by using a three level method based on non-polynomial quintic spline in space and finite difference discretization in time. We also obtain two new high accuracy schemes of O (k⁴, h⁶) and O (k⁴, h⁸) and two new schemes which are analogues of Jain's formula for the non-homogeneous case. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

AMLI preconditioner for the p‐version of the FEM

From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p‐version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix K_h,k resulting from the h‐version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner C_h,k for the matrix K_h,k and show that the condition number of C K_h,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on quadrilaterals

Numerical simulation of industrial processes involving viscoelastic liquids is often based on finite element methods on quadrilateral meshes. However, numerical analysis of these methods has so far been limited to triangular meshes. In this work, we consider quadrilateral meshes. We first study the approximation of the transport equation by a Galerkin discontinuous method and prove an 𝒪(h^k+1/2) error estimates for the Q_k finite element. Then we study a differential model for viscoelastic flow with unknowns u the velocity, p the pressure, and σ the viscoelastic part of the extra-stress tensor. The approximations are ((Q₁)² transforms of) Q_k+1 continuous for u, Q_k discontinuous for σ, and P_k discontinuous for p, with k ≥ 1. Upwinding for σ is obtained by the Galerkin discontinuous method. We show that an error estimate of order 𝒪(h^k+1/2) is valid in the energy norm for the three unknowns. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 97–114, 1998     

Numerical study using explicit multistep Galerkin finite element method for the MRLW equation

In this article, an explicit multistep Galerkin finite element method for the modified regularized long wave equation is studied. The discretization of this equation in space is by linear finite elements, and the time discretization is based on explicit multistep schemes. Stability analysis and error estimates of our numerical scheme are derived. Numerical experiments indicate the validation of the scheme by L₂– and L_∞– error norms and three invariants of motion.4 © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1875–1889, 2015     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs

We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal domains by the boundary concentrated finite element method. We prove that the discretization error ||u^*-u_h^*||_L²(G)\|u^{*}-u_{h}^{*}\|_{L^{2}(\Gamma)} decreases like N ⁻¹, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the discretization error behaves like N ^−3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The paper is complemented with numerical results.     

Neumann‐Neumann methods for a DG discretization on geometrically nonconforming substructures

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ω_i of size O(H_i). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ω_i, a conforming finite element space associated to a triangulation \begin{align*} {\mathcal{T}}_{h_i}(\Omega_i)\end{align*} is introduced. To handle the nonmatching meshes across ?Ω_i, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ?Ω_i, a condition number estimate \begin{align*} C(1 + \max_i\log \frac{H_i}{h_i})^2\end{align*} is established with C independent of h_i, H_i, h_i/h_j, and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Convergence of Exponential Attractors for a Finite Element Approximation of the Allen–Cahn Equation

Abstract

We consider a space semidiscretization of the Allen–Cahn equation by continuous piecewise linear finite elements. For every mesh parameter h, we build an exponential attractor of the dynamical system associated with the approximate equations. We prove that, as h tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated with the Allen–Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semigroup. Here, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.     

Second‐order finite‐volume schemes for a non‐linear hyperbolic equation: error estimate

We study second‐order finite‐volume schemes for the non‐linear hyperbolic equation u_t(x, t) + div F(x, t, u(x, t)) = 0 with initial condition u₀. The main result is the error estimate between the approximate solution given by the scheme and the entropy solution. It is based on some stability properties verified by the scheme and on a discrete entropy inequality. If u₀ ∈ L^∞ ∩ BV_loc(ℝ^N), we get an error estimate of order h^1/4, where h defines the size of the mesh. Copyright © 2000 John Wiley & Sons, Ltd.     

Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h ² max (1,logk ^− 1). Some simple numerical examples illustrate this convergence behaviour in practice. We thank the University of New South Wales for financial support provided by a Faculty Research Grant.     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010