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CONVERGENCE ANALYSIS FOR FINITE ELEMENT DISCRETIZATIONS OF THE HELMHOLTZ EQUATION WITH DIRICHLETTONEUMANN BOUNDARY CONDITIONS
"... Abstract. A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R d, d ∈{1, 2, 3} is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasioptimality of the method. As an application of the general theory, ..."
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Cited by 21 (7 self)
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Abstract. A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R d, d ∈{1, 2, 3} is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasioptimality of the method. As an application of the general theory, a full error analysis of the classical hpversion of the finite element method (hpFEM) is presented for the model problem where the dependence on the mesh width h, the approximation order p, andthewavenumberk is given explicitly. In particular, it is shown that quasioptimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k). 1.
WaveNumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation
 SIAM J. Numer. Anal
"... In this paper, we develop a new stability and convergence theory for highly indefinite elliptic partial differential equations by considering the Helmholtz equation at high wave number as our model problem. The key element in this theory is a novel kexplicit regularity theory for Helmholtz boundary ..."
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Cited by 20 (7 self)
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In this paper, we develop a new stability and convergence theory for highly indefinite elliptic partial differential equations by considering the Helmholtz equation at high wave number as our model problem. The key element in this theory is a novel kexplicit regularity theory for Helmholtz boundary value problems that is based on decomposing the solution into in two parts: the first part has the H2Sobolev regularity expected of elliptic PDEs but features kindependent regularity constants; the second part is an analytic function for which kexplicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of boundary value problems including the case Robin boundary conditions in domains with analytic boundary and in convex polygons. As the most important practical application we apply our full error analysis to the classical hpversion of the finite element method (hpFEM) where the dependence on the mesh width h, the approximation order p, and the wave number k is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems grows only polynomially in k, it is shown that quasioptimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k).
A class of discontinuous Petrov–Galerkin methods. Part IV: the optimal test norm and timeharmonic wave propagation in 1D
, 2011
"... Abstract The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the highfrequency range. This paper presents a new method with no phase errors for onedimensional (1D) timehar ..."
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Abstract The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the highfrequency range. This paper presents a new method with no phase errors for onedimensional (1D) timeharmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the Discontinuous PetrovGalerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L 2 norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.
Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number
 Mathematics of Computation
"... Abstract. Two local discontinuous Galerkin (LDG) methods using some nonstandard numerical fluxes are developed for the Helmholtz equation with the first order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are absolutely stable (hence wellposed ..."
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Abstract. Two local discontinuous Galerkin (LDG) methods using some nonstandard numerical fluxes are developed for the Helmholtz equation with the first order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are absolutely stable (hence wellposed) with respect to both the wave number and the mesh size. Optimal order (with respect to the mesh size) error estimates are proved for all wave numbers in the preasymptotic regime. To analyze the proposed LDG methods, they are recasted and treated as (nonconforming) mixed finite element methods. The crux of the analysis is to establish a generalized infsup condition, which holds without any mesh constraint, for each LDG method. The generalized infsup conditions then easily infer the desired absolute stability of the proposed LDG methods. In return, the stability results not only guarantee the wellposedness of the LDG methods but also play a crucial role in the derivation of the error estimates. Numerical experiments, which confirm the theoretical results and compare the proposed two LDG methods, are also presented in the paper. 1.
AN EXPONENTIALLY CONVERGENT NONPOLYNOMIAL FINITE ELEMENT METHOD FOR TIMEHARMONIC SCATTERING FROM POLYGONS
"... Abstract. In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We pr ..."
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Cited by 7 (3 self)
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Abstract. In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of twodimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners, and representing the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a leastsquares functional, which we discretize in such a way that a matrix leastsquares problem is obtained. We give computable exponential bounds on the rate of convergence of the leastsquares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.
HYBRIDIZING RAVIARTTHOMAS ELEMENTS FOR THE HELMHOLTZ EQUATION
"... Abstract. This paper deals with the application of hybridized mixed methods for discretizing the Helmholtz problem. Starting from a mixed formulation, where the flux is considered a separate unknown, we use RaviartThomas finite elements to approximate the solution. We present two ways of hybridizin ..."
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Abstract. This paper deals with the application of hybridized mixed methods for discretizing the Helmholtz problem. Starting from a mixed formulation, where the flux is considered a separate unknown, we use RaviartThomas finite elements to approximate the solution. We present two ways of hybridizing the problem, which means breaking the normal continuity of the fluxes and then imposing continuity weakly via functions supported on the element faces or edges. The first method is the UltraWeak Variational Formulation, first introduced by Cessenat and Després [7]; the second one uses Lagrange multipliers on element interfaces. We compare the two methods, and give numerical results. We observe that the iterative solvers applied to the two methods behave well for large wave numbers. 1.
Generalized DGMethods for Highly Indefinite Helmholtz Problems based on the UltraWeak Variational Formulation
"... We develop a stability and convergence theory for the Ultra Weak Variational Formulation (UWVF) of a highly indefinite Helmholtz problem in R d, d ∈ {1, 2, 3}. The theory covers conforming as well as nonconforming generalized finite element methods. In contrast to conventional Galerkin methods where ..."
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Cited by 4 (3 self)
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We develop a stability and convergence theory for the Ultra Weak Variational Formulation (UWVF) of a highly indefinite Helmholtz problem in R d, d ∈ {1, 2, 3}. The theory covers conforming as well as nonconforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the UWVF admits a unique solution under much weaker conditions. As an application we present the error analysis for the hpversion of the finite element method explicitly in terms of the mesh width h, polynomial degree p and wave number k. It is shown that the optimal convergence order estimate is obtained under the conditions that kh / √ p is sufficiently small and the polynomial degree p is at least O(log k). AMS Subject Classifications: 35J05, 65N12, 65N30 Key words: Helmholtz equation at high wavenumber, stability, convergence, discontinuous
Error analysis of Trefftzdiscontinuous Galerkin methods for the timeharmonic Maxwell equations
 Math. Comp
, 2013
"... Abstract. In this paper, we extend to the timeharmonic Maxwell equations the p–version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the pversion, SIAM J. Numer. Anal., 49 (2011), 264284 ..."
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Abstract. In this paper, we extend to the timeharmonic Maxwell equations the p–version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the pversion, SIAM J. Numer. Anal., 49 (2011), 264284] for Trefftzdiscontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a meshskeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a meshindependent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vectorvalued plane wave functions is considered, and convergence rates are derived. 1.
On the negativeorder norm accuracy of a localstructurepreserving LDG method
 J. Sci. Comput
"... Abstract The accuracy in negativeorder norms is examined for a localstructurepreserving local discontinuous Galerkin method for the Laplace equation [Li and Shu, Methods and Applications of Analysis, v13 (2006), pp.215233]. With its distinctive feature in using harmonic polynomials as local app ..."
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Abstract The accuracy in negativeorder norms is examined for a localstructurepreserving local discontinuous Galerkin method for the Laplace equation [Li and Shu, Methods and Applications of Analysis, v13 (2006), pp.215233]. With its distinctive feature in using harmonic polynomials as local approximating functions, this method has lower computational complexity than the standard local discontinuous Galerkin method while keeping the same order of accuracy in both the energy and the L 2 norms. In this note, numerical experiments are presented to demonstrate some accuracy loss of the method in negativeorder norms.