arXiv subjects

Haijun Wu

Publications and source records attributed to Haijun Wu.

48 records · Page 3Linked to original sources

arXiv2017

Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number

We study superconvergence property of the linear finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The $H^1$-error estimate with explicit dependence on the wave number $k$ {is} derived. First, we prove that under the assumption $k(kh)^2\leq C_0$ ($h$ is the mesh size) and certain mesh condition, the estimate between the finite element solution and the linear interpolation of the exact solution is superconvergent under the $H^1$-seminorm, although the pollution error still exists. Second, we prove a similar result for the recovered gradient by PPR and found that the PPR can only improve the interpolation error and has no effect on the pollution error. Furthermore, we estimate the error between the finite element gradient and recovered gradient and discovered that the pollution error is canceled between these two quantities. Finally, we apply the Richardson extrapolation to recovered gradient and demonstrate numerically that PPR combined with the Richardson extrapolation can reduce the interpolation and pollution errors simultaneously, and therefore, leads to an asymptotically exact {\it a posteriori} error estimator. All theoretical findings are verified by numerical tests.

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arXiv2014

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 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, $p$ 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 great capability of the CIP-FEM in reducing the pollution effect.

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arXiv2013

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}.

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arXiv2012

An absolutely stable discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations with large wave number

This paper develops and analyzes an interior penalty discontinuous Galerkin (IPDG) method using piecewise linear polynomials for the indefinite time harmonic Maxwell equations with the impedance boundary condition in the three dimensional space. The main novelties of the proposed IPDG method include the following: first, the method penalizes not only the jumps of the tangential component of the electric field across the element faces but also the jumps of the tangential component of its vorticity field; second, the penalty parameters are taken as complex numbers of negative imaginary parts. For the differential problem, we prove that the sesquilinear form associated with the Maxwell problem satisfies a generalized weak stability (i.e., inf-sup condition) for star-shaped domains.Such a generalized weak stability readily infers wave-number explicit a priori estimates for the solution of the Maxwell problem, which plays an important role in the error analysis for the IPDG method. For the proposed IPDG method, we show that the discrete sesquilinear form satisfies a coercivity for all positive mesh size $h$ and wave number $k$ and for general domains including non-star-shaped ones. In turn, the coercivity easily yields the well-posedness and stability estimates (i.e., a priori estimates) for the discrete problem without imposing any mesh constraint. Based on these discrete stability estimates, by adapting a nonstandard error estimate technique of Fung and Wu (2009), we derive both the energy-norm and the $L^2$-norm error estimates for the IPDG method in all mesh parameter regimes including pre-asymptotic regime (i.e., $k^2 h\gtrsim 1$). Numerical experiments are also presented to gauge the theoretical results and to numerically examine the pollution effect (with respect to $k$) in the error bounds.

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arXiv2012

A combined finite element and multiscale finite element method for the multiscale elliptic problems

The oversampling multiscale finite element method (MsFEM) is one of the most popular methods for simulating composite materials and flows in porous media which may have many scales. But the method may be inapplicable or inefficient in some portions of the computational domain, e.g., near the domain boundary or near long narrow channels inside the domain due to the lack of permeability information outside of the domain or the fact that the high-conductivity features cannot be localized within a coarse-grid block. In this paper we develop a combined finite element and multiscale finite element method (FE-MsFEM), which deals with such portions by using the standard finite element method on a fine mesh and the other portions by the oversampling MsFEM. The transmission conditions across the FE-MSFE interface is treated by the penalty technique. A rigorous convergence analysis for this special FE-MsFEM is given under the assumption that the diffusion coefficient is periodic. Numerical experiments are carried out for the elliptic equations with periodic and random highly oscillating coefficients, as well as multiscale problems with high contrast channels, to demonstrate the accuracy and efficiency of the proposed method.

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arXiv2012

Continuous Interior Penalty Finite Element Method for Helmholtz Equation with High Wave Number: One Dimensional Analysis

This paper 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 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 $H^1$-norm, as the sum of best approximation plus a pollution term that is the order of the phase difference. It is proved that the pollution 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.

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arXiv2012

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

In this paper, which is part II in a series of two, the pre-asymptotic 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, pre-asymptotic error estimates are derived for both methods under the condition of $\frac{kh}{p}\le C_0\big(\frac{p}{k}\big)^{\frac{1}{p+1}}$, 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 $O\Big(\frac{k}{p^2}\big(\frac{kh}{σp}\big)^{2p}\Big)$ if the exact solution $u\in H^2(\Om)$ which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here $\si$ is a constant independent of $k, h, p$, and the penalty parameters. Moreover, it is proved that the CIP-FEM is stable for any $k, h, p>0$ and penalty parameters with positive imaginary parts. Besides the advantage of the absolute stability of the CIP-FEM compared to the FEM, the penalty parameters may be tuned to reduce the pollution effects.

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arXiv2011

Continuous Interior Penalty Finite Element Methods for the Helmholtz Equation with Large Wave Number

This paper develops and analyzes some continuous interior penalty finite element methods (CIP-FEMs) using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions. The novelty of the proposed methods is to use complex penalty parameters with positive imaginary parts. It is proved that, if the penalty parameter is a pure imaginary number $ı\ga$ with $0<\ga\le C$, then the proposed CIP-FEM is stable (hence well-posed) without any mesh constraint. Moreover the method satisfies the error estimates $C_1kh+C_2k^3h^2$ in the $H^1$-norm when $k^3h^2\le C_0$ and $C_1kh+\frac{C_2}{\ga}$ when $k^3h^2> C_0$ and $kh$ is bounded, where $k$ is the wave number, $h$ is the mesh size, and the $C$'s are positive constants independent of $k$, $h$, and $\ga$. Optimal order $L^2$ error estimates are also derived. The analysis is also applied if the penalty parameter is a complex number with positive imaginary part. By taking $\ga\to 0+$, the above estimates are extended to the linear finite element method under the condition $k^3h^2\le C_0$. Numerical results are provided to verify the theoretical findings. It is shown that the penalty parameters may be tuned to greatly reduce the pollution errors.

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arXiv2010

An unfitted $hp$-interface penalty finite element method for elliptic interface problems

An $hp$ version of interface penalty finite element method ($hp$-IPFEM) is proposed for elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken $H^1$ norm, which are optimal with respect to $h$ and suboptimal with respect to $p$ by half an order of $p$, are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in $L^2$ norm are proved by the duality argument.

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arXiv2009

$hp$-discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

This paper develops some interior penalty $hp$-discontinuous Galerkin ($hp$-DG) methods for the Helmholtz equation in two and three dimensions. The proposed $hp$-DG methods are defined using a sesquilinear form which is not only mesh-dependent but also degree-dependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order $p$. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed $hp$-discontinuous Galerkin methods are absolutely stable (hence, well-posed). For each fixed wave number $k$, sub-optimal order error estimates in the broken $H^1$-norm and the $L^2$-norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition $k^3h^2p^{-1}\le C_0$ by utilizing these stability and error estimates and using a stability-error iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}, which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size $h$, the polynomial degree $p$, the wave number $k$, as well as all the penalty parameters for the numerical solutions.

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arXiv2008

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and sub-optimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}.

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arXiv2007

A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+\De\bigl(\eps \De u-\eps^{-1} f(u)\bigr)=0$. It is shown that the {\it a posteriori} error bounds depends on $\eps^{-1}$ only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

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