Interior Penalty Finite Element Method 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.

Numerical solution of the cavity scattering problem for flexural waves on thin plates: Linear finite element methods

Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.

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.

A C0 interior penalty method for the phase field crystal equation

AbstractWe present a C0 interior penalty finite element method for the sixth‐order phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C0 interior penalty finite element method developed for the phase field crystal equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C0 interior penalty method. We furthermore benchmark our method against numerical experiments previously established in the literature.

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.

Goal-oriented a posteriori error estimation for the biharmonic problem based on an equilibrated moment tensor

In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment tensor. Then, an abstract unified framework for the goal-oriented a posteriori error estimation is derived based on the equilibrated moment tensor and the potential reconstruction that provides a guaranteed upper bound for the error of a numerical approximation for the goal functional. In particular, C0 interior penalty and discontinuous Galerkin finite element methods are employed for the practical realisation of the estimators. Numerical experiments are performed to illustrate the effectivity of the estimators.

Adaptive configurational force‐based propagation for brittle and fatigue crack analysis

AbstractThis article presents a ‐adaptive crack propagation method for highly accurate 2D crack propagation paths which requires no a priori knowledge of the tip solution. The propagation method is designed to be simple to implement, only ‐adaptivity is required, with the propagation step size independent of the initial mesh allowing users to obtain high fidelity crack path predictions for domains containing multiple cracks propagating at different rates. The proposed method also includes a crack path derefinement scheme, where elements away from the crack tip are derefined whilst elements close to the crack tip are small so capture the fidelity of the crack path. The result is that the propagation of cracks over an increasingly larger distances has negligible increased computational effort and effect on the propagation path prediction. The linear elastic problem is solved using the hp discontinuous Galerkin symmetric interior penalty finite element method, which is post‐processed to obtain the configurational force at each tip to a user defined accuracy. Several numerical examples are used to demonstrate the accuracy, efficiency, and capability of the method. Due to the method's high accuracy crack path solutions of benchmark problems that are prolifically used in the literature are challenged.

Convergence of Adaptive Discontinuous Galerkin and $$C^0$$-Interior Penalty Finite Element Methods for Hamilton–Jacobi–Bellman and Isaacs Equations

We prove the convergence of adaptive discontinuous Galerkin and \(C^0\)-interior penalty methods for fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including Poincaré and trace inequalities, a proof of the density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results.

Unified analysis of discontinuous Galerkin andC0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

We provide a unified analysis ofa posteriorianda priorierror bounds for a broad class of discontinuous Galerkin andC0-IP finite element approximations of fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions inH2of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for thea priorierror analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda–Talenti inequality to piecewise polynomial vector fields.

A C0 interior penalty method for the Cahn-Hilliard equation

We present a C \begin{document}$ ^0 $\end{document} interior penalty method for the Cahn-Hilliard equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C \begin{document}$ ^0 $\end{document} interior penalty finite element method developed for the Cahn-Hilliard equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C \begin{document}$ ^0 $\end{document} interior penalty method. We furthermore support our conclusions with a few numerical experiments.

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.

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

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.

Formula omitted] interior penalty methods for a dynamic nonlinear beam model

In this work, we aim to develop efficient numerical schemes for a nonlinear fourth-order partial differential equation arising from the so-called dynamic Gao beam model. We use C0 interior penalty finite element methods over the spatial domain to set up the semi-discrete formulations. Convergence results for the semi-discrete case are shown, based on a truncated variational formulation and its equivalent abstract formulations. We combine time discretizations to derive fully discrete numerical formulations. Newton’s method is applied to compute one time step numerical solutions of a nonlinear system. Two numerical examples are provided: one supports our theoretical results and the other presents a buckling state of the Gao beams.

Skeleton-stabilized IsoGeometric Analysis: High-regularity interior-penalty methods for incompressible viscous flow problems

A Skeleton-stabilized IsoGeometric Analysis (SIGA) technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed method allows utilizing identical finite dimensional spaces (with arbitrary B-splines/NURBS order and regularity) for the approximation of the pressure and velocity components. The key idea is to stabilize the jumps of high-order derivatives of variables over the skeleton of the mesh. For B-splines/NURBS basis functions of degree k with Cα-regularity (0≤α<k), only the derivative of order α+1 has to be controlled. This stabilization technique thus can be viewed as a high-regularity generalization of the (Continuous) Interior-Penalty Finite Element Method. Numerical experiments are performed for the Stokes and Navier–Stokes equations in two and three dimensions. Oscillation-free solutions and optimal convergence rates are obtained. In terms of the sparsity pattern of the algebraic system, we demonstrate that the block matrix associated with the stabilization term has a considerably smaller bandwidth when using B-splines than when using Lagrange basis functions, even in the case of C0-continuity. This important property makes the proposed isogeometric framework practical from a computational effort point of view.

Pin-wise homogenization for SPN neutron transport approximation using the finite element method

The neutron transport equation describes the distribution of neutrons inside a nuclear reactor core. Homogenization strategies have been used for decades to reduce the spatial and angular domain complexity of a nuclear reactor by replacing previously calculated heterogeneous subdomains by homogeneous ones and using a low order transport approximation to solve the new problem. The generalized equivalence theory for homogenization looks for discontinuous solutions through the introduction of discontinuity factors at the boundaries of the homogenized subdomains. In this work, the generalized equivalence theory is extended to the Simplified P N equations using the finite element method. This extension proposes pin discontinuity factors instead of the usual assembly discontinuity factors and the use of the simplified spherical harmonics approximation rather than diffusion theory. An interior penalty finite element method is used to discretize and solve the problem using discontinuity factors. One dimensional numerical results show that the proposed pin discontinuity factors produce more accurate results than the usual assembly discontinuity factors. The proposed pin discontinuity factors produce precise results for both pin and assembly averaged values without using advanced reconstruction methods. Also, the homogenization methodology is verified against the calculation performed with reference discontinuity factors.

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.