Continuous Interior Penalty Finite Element Research Articles

An optimized CIP-FEM to reduce the pollution errors for the Helmholtz equation on a general unstructured mesh

The continuous interior penalty finite element method (CIP-FEM) has shown promise in reducing pollution errors when numerically simulating the Helmholtz equation with high wave numbers. However, its reliance on structured meshes largely limits its applicability. To address this limitation, this paper presents a novel approach to CIP-FEM on general unstructured meshes. The interior penalty parameters of the CIP-FEM are determined in an offline phase through minimizing the residual obtained by substituting the plane waves into the CIP finite element equation. To enhance accuracy, a quasi-Newton algorithm is employed to correct the penalty parameters and minimize errors of the CIP finite element approximations to the plane waves. The calculated interior penalty parameters can be saved, enabling the online solutions of CIP-FEM to be obtained by reusing these parameters for arbitrary sources during practical computations. Numerical experiments are presented to demonstrate the significant reduction of pollution errors for both linear and higher-order CIP-FEMs on unstructured meshes. Furthermore, the proposed algorithm successfully simulates high-frequency acoustic fields in a three-dimensional vehicle cabin with greatly reduced errors within certain fixed computational time, demonstrating its practical effectiveness in real-world scenarios.

Dispersion Analysis of CIP-FEM for the Helmholtz Equation

When solving the Helmholtz equation numerically, the accuracy of numerical solution deteriorates as the wave number increases, which is known as “pollution effect” and which is directly related to the phase difference between the exact and numerical solutions, caused by the numerical dispersion. In this paper, we propose a dispersion analysis for the continuous interior penalty finite element method (CIP-FEM) and derive an explicit formula of the penalty parameter for the th order CIP-FEM on tensor product (Cartesian) meshes, with which the phase difference is reduced from to . Extensive numerical tests show that the pollution error of the CIP-FE solution is also reduced by two orders in with the same penalty parameter.

Adaptive FEM for Helmholtz Equation with Large Wavenumber

A posteriori upper and lower bounds are derived for the linear finite element method (FEM) for the Helmholtz equation with large wavenumber. It is proved rigorously that the standard residual type error estimator seriously underestimates the true error of the FE solution for the mesh size h in the preasymptotic regime, which is first observed by Babuška et al. (Int J Numer Methods Eng 40:3443–3462, 1997) for a one dimensional problem. By establishing an equivalence relationship between the error estimators for the FE solution and the corresponding elliptic projection of the exact solution, an adaptive algorithm is proposed and its convergence and quasi-optimality are proved under the condition that \(k^3h_0^{1+\alpha }\) is sufficiently small, where k is the wavenumber, \(h_0\) is the initial mesh size, and \(\frac{1}{2}<\alpha \le 1\) is a regularity constant depending on the maximum reentrant angle of the domain. Numerical tests are given to verify the theoretical findings and to show that the adaptive continuous interior penalty finite element method (CIP-FEM) with appropriately selected penalty parameters can greatly reduce the pollution error and hence the residual type error estimator for this CIP-FEM is reliable and efficient even in the preasymptotic regime.

Continuous interior penalty finite element methods for the time-harmonic Maxwell equation with high wave number

In this paper, using the first-order Nedelec conforming edge element space of the second type, we develop and analyze a continuous interior penalty finite element method (CIP-FEM) for the time-harmonic Maxwell equation in the three-dimensional space. Compared with the standard finite element methods, the novelty of the proposed method is that we penalize the jumps of the tangential component of its vorticity field. It is proved that if the penalty parameter is a complex number with negative imaginary part, then the CIP-FEM is well-posed without any mesh constraint. The error estimates for the CIP-FEM are derived. Numerical experiments are presented to verify our theoretical results.

A Robust Multilevel Preconditioner Based on a Domain Decomposition Method for the Helmholtz Equation

In this paper, we present a robust multilevel preconditioner for the algebraic system resulting from the continuous interior penalty finite element method for the approximation of the Helmholtz equation. The key idea in this work is the replacement of traditional smoothers by the one level overlapping domain decomposition method on coarse grids. The proposed multilevel method then serves as a preconditioner in the outer GMRES iteration. Numerical results show that for fixed wave numbers, the convergence of our multilevel method is independent of the mesh size. Furthermore, the performance of the algorithm depends relatively mildly on the wave number.

An Adaptive CIP-FEM for the Polygonal-Line Grating Problem

This paper is concerned with the diffraction by a polygonal-line grating. We develop a continuous interior penalty finite element method based on the truncation of the nonlocal boundary operators for solving the problem. An a posteriori error estimate is derived for the method. The truncation parameter is determined through the truncation error of the a posteriori error estimate. Numerical experiments are also presented to show the efficiency and robustness of the proposed adaptive algorithm.

Convergence and Quasi-Optimality of an Adaptive Continuous Interior Penalty Finite Element Method

Convergence and Quasi-Optimality of an Adaptive Continuous Interior Penalty Finite Element Method

Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations

AbstractThis paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lowerH1+sweak regularity under consideration, where 0 ≤s≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliaryC1finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem

We propose a continuous interior penalty finite element method designed for a third-order singularly perturbed problem. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Moreover, we show numerical experiments which support our theoretical findings.

Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

This article addresses the properties of continuous interior penalty (CIP) finite element solutions for the Helmholtz equation. The h-version of the CIP finite element method with piecewise linear approximation is applied to a one-dimensional (1D) model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the H1-norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in the 1D case also works very well for the CIP-FEM on two-dimensional Cartesian grids.

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}.

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

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.