893 research outputs found
-discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number
This paper develops some interior penalty -discontinuous Galerkin
(-DG) methods for the Helmholtz equation in two and three dimensions. The
proposed -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
. Furthermore, to ensure the stability, the penalty parameters are taken as
complex numbers with positive imaginary parts. It is proved that the proposed
-discontinuous Galerkin methods are absolutely stable (hence, well-posed).
For each fixed wave number , sub-optimal order error estimates in the broken
-norm and the -norm are derived without any mesh constraint. The
error estimates and the stability estimates are improved to optimal order under
the mesh condition 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
, the polynomial degree , the wave number , as well as all the penalty
parameters for the numerical solutions.Comment: 27 page
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
A pure source transfer domain decomposition method for Helmholtz equations in unbounded domain
We propose a pure source transfer domain decomposition method (PSTDDM) for
solving the truncated perfectly matched layer (PML) approximation in bounded
domain of Helmholtz scattering problem. The method is a modification of the
STDDM proposed by [Z. Chen and X. Xiang, SIAM J. Numer. Anal., 51 (2013), pp.
2331--2356]. After decomposing the domain into non-overlapping layers, the
STDDM is composed of two series steps of sources transfers and wave expansions,
where truncated PML problems on two adjacent layers and truncated
half-space PML problems are solved successively. While the PSTDDM consists
merely of two parallel source transfer steps in two opposite directions, and in
each step truncated PML problems on two adjacent layers are solved
successively. One benefit of such a modification is that the truncated PML
problems on two adjacent layers can be further solved by the PSTDDM along
directions parallel to the layers. And therefore, we obtain a block-wise PSTDDM
on the decomposition composed of squares, which reduces the size of
subdomain problems and is more suitable for large-scale problems. Convergences
of both the layer-wise PSTDDM and the block-wise PSTDDM are proved for the case
of constant wave number. Numerical examples are included to show that the
PSTDDM gives good approximations to the discrete Helmholtz equations with
constant wave numbers and can be used as an efficient preconditioner in the
preconditioned GMRES method for solving the discrete Helmholtz equations with
constant and heterogeneous wave numbers.Comment: 31 pages, 7 figure
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.Comment: 29 pages and 7 figure
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 and wave number
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
-norm error estimates for the IPDG method in all mesh parameter regimes
including pre-asymptotic regime (i.e., ). Numerical experiments
are also presented to gauge the theoretical results and to numerically examine
the pollution effect (with respect to ) in the error bounds.Comment: 11 figures and 1 tabl
An unfitted -interface penalty finite element method for elliptic interface problems
An version of interface penalty finite element method (-IPFEM) is
proposed for elliptic interface problems in two and three dimensions on
unfitted meshes. Error estimates in broken norm, which are optimal with
respect to and suboptimal with respect to by half an order of , are
derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates
in norm are proved by the duality argument
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. - and - error
estimates with explicit dependence on the wave number are derived. In
particular, it is shown that if is sufficiently small, then
the pollution errors of both methods in -norm are bounded by
, which coincides with the phase error of the FEM obtained
by existent dispersion analyses on Cartesian grids, where is the mesh size,
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 -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
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 -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
-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
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 -error estimate with explicit
dependence on the wave number {is} derived.
First, we prove that under the assumption ( 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 -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.Comment: 25 pages, 16 figures. arXiv admin note: substantial text overlap with
arXiv:1612.0338
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 \i\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 in the
-norm when and C_1kh+\frac{C_2}{\ga} when
and is bounded, where is the wave number, is the mesh size, and
the 's are positive constants independent of , , and \ga. Optimal
order 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 . 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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