Interior Penalty Finite Element Method Research Articles

A Robust Multilevel Method for the Time-harmonic Maxwell Equation with High Wave Number

A robust multilevel preconditioner for the time-harmonic Maxwell equation with high wave number is presented in this paper. We choose a stable continuous interior penalty finite element method as a coarse correction space and design different smoothers for the indefinite algebraic system in different mesh levels. The multilevel method is performed as a preconditioner in the outer GMRES iteration. To give quantitative insight of our algorithm we use local Fourier analysis to analyze the convergence property of the proposed multilevel method. Numerical results are presented to confirm the efficiency of our multilevel method.

Convergence analysis of an adaptive continuous interior penalty finite element method for the Helmholtz equation

We are concerned with an adaptive continuous interior penalty finite element method for the Helmholtz equation. A convergence result with explicit constraints on the wave number k and the initial mesh size h0 is derived. In particular, it is shown that if k1+sh0s is sufficiently small, where 12<s≤1 depends on the regularity of the weak solution of the model problem, then the adaptive continuous interior penalty finite element method is a contraction with respect to the sum of the energy error and a scaled error estimator. Numerical experiments are provided to verify the theoretical findings and show advantages of the adaptive continuous interior penalty finite element method.

Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number

A preasymptotic error analysis of the finite element method (FEM) and some continuous interior penalty finite element method (CIP-FEM) for the Helmholtz equation in two and three dimensions is proposed. $H^1$- and $L^2$-error estimates with explicit dependence on the wave number $k$ are derived. In particular, it is shown that if $k^{2p+1}h^{2p}$ is sufficiently small, then the pollution errors of both methods in $H^1$-norm are bounded by $O(k^{2p+1}h^{2p})$, which coincides with the phase error of the FEM obtained by existent dispersion analyses on Cartesian grids, where $h$ is the mesh size, and $p$ is the order of the approximation space and is fixed. The CIP-FEM extends the classical one by adding more penalty terms on jumps of higher (up to $p$th order) normal derivatives in order to reduce efficiently the pollution errors of higher order methods. Numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the CIP-FEM in reducing the pollution effect.

Multilevel Preconditioner with Stable Coarse Grid Corrections for the Helmholtz Equation

In this paper we consider a class of robust multilevel precontioners for the Helmholtz equation with high wave number. The key idea in this work is to use the continuous interior penalty finite element methods (CIP-FEM) studied in \cite{Wu12,Wu12-hp} to construct the stable coarse grid correction problems. The multilevel methods, based on GMRES smoothing on coarse grids, are then served as a preconditioner in the outer GMRES iteration. In the one dimensional case, convergence property of the modified multilevel methods is analyzed by the local Fourier analysis. From our numerical results, we find that the proposed methods are efficient for a reasonable range of frequencies. The performance of the algorithms depends relatively mildly on wave number. In particular, only one GMRES smoothing step may guarantee the optimal convergence of our multilevel algorithm, which remedies the shortcoming of the multilevel algorithm in \cite{EEO01}.

A fully discrete [formula omitted] interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equation

A fully discrete C0 interior penalty finite element method is proposed and analyzed for the Extended Fisher–Kolmogorov (EFK) equation ut+γΔ2u−Δu+u3−u=0 with appropriate initial and boundary conditions, where γ is a positive constant. We derive a regularity estimate for the solution u of the EFK equation that is explicit in γ and as a consequence we derive a priori error estimates that are robust in γ.

Preasymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: $hp$ Version

In this paper, which is the second in a series of two, the preasymptotic error analysis of the continuous interior penalty finite element method (CIP-FEM) and the FEM for the Helmholtz equation in two and three dimensions is continued. While Part I contained results on the linear CIP-FEM and FEM, the present part deals with approximation spaces of order $p \ge 1$. By using a modified duality argument, preasymptotic error estimates are derived for both methods under the condition of $\frac{kh}{p}\le C_0(\frac{p}{k})^{\frac{1}{p+1}}$, where $k$ is the wave number, $h$ is the mesh size, and $C_0$ is a constant independent of $k, h, p$, and the penalty parameters. It is shown that the pollution errors of both methods in $H^1$-norm are $O(k^{2p+1}h^{2p})$ if $p=O(1)$ and are $O(\frac{k}{p^2}(\frac{kh}{\sigma p})^{2p})$ if the exact solution $u\in H^2(\Omega)$ which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here $\sigma$ is a constant independent of $k, h, p$ and the penalty par...

Continuous interior penalty finite element methods for Sobolev equations with convection‐dominated term

AbstractWe consider implicit and semi‐implicit time‐stepping methods for continuous interior penalty (CIP) finite element approximations of Sobolev equations with convection‐dominated term. Stability is obtained by adding an interior penalty term giving L2 ‐control of the jump of the gradient over element faces. Several $\cal {A}$ ‐stable time‐stepping methods are analyzed and shown to be unconditionally stable and optimally convergent. We show that the contribution from the gradient jumps leading to an extended matrix pattern may be extrapolated from previous time steps, and hence handled explicitly without loss of stability and accuracy. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

On pressure separation algorithms (PSepA) for improving the accuracy of incompressible flow simulations

AbstractWe investigate a special technique called ‘pressure separation algorithm’ (PSepA) (see Applied Mathematics and Computation 2005; 165:275–290 for an introduction) that is able to significantly improve the accuracy of incompressible flow simulations for problems with large pressure gradients. In our numerical studies with the computational fluid dynamics package FEATFLOW (www.featflow.de), we mainly focus on low‐order Stokes elements with nonconforming finite element approximations for the velocity and piecewise constant pressure functions. However, preliminary numerical tests show that this advantageous behavior can also be obtained for higher‐order discretizations, for instance, with Q2/P1 finite elements. We analyze the application of this simple, but very efficient, algorithm to several stationary and nonstationary benchmark configurations in 2D and 3D (driven cavity and flow around obstacles), and we also demonstrate its effect to spurious velocities in multiphase flow simulations (‘static bubble’ configuration) if combined with edge‐oriented, resp., interior penalty finite element method stabilization techniques. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Interior penalty variational multiscale method for the incompressible Navier–Stokes equation: Monitoring artificial dissipation

In this paper, we apply the interior penalty method analyzed in Burman and Fernàndez [E. Burman, M. Fernàndez, Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math. (2007)] to flows at high Reynolds number. As a possible measure of solution quality we propose to monitor the ratio between the artificial dissipation induced by the numerical method and the computed physical dissipation. For smooth flows we prove that for our method the artificial dissipation serves as an a posteriori error estimator and also that it vanishes at an optimal rate. In the case of flows with multiscale features, we discuss a heuristic approach relating the decay of the artificial dissipation to the decay rate of the power spectrum. Some numerical results in two space dimensions are presented examining the relation between the numerical dissipation and solution quality.

Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence

This paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible Navier–Stokes equations. Incompressibility and convective effects are both stabilized adding an interior penalty term giving L 2-control of the jump of the gradient of the approximate solution over the internal faces. Using continuous equal-order finite elements for both velocities and pressures, in a space semi-discretized formulation, we prove convergence of the approximate solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations, provided the exact solution is smooth.

Continuous Interior Penalty Finite Element Method for Oseen's Equations

In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207-216] to Oseen's equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L2 -control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.

An Interior Penalty Finite Element Method with Discontinuous Elements

A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and $L^2$-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.