Chapter 6: Overview of Variational Formulations for Linear Elliptic PDEs

Spence, Euan A.

doi:10.1137/1.9781611973822.ch6

Cited by 14 publications

(16 citation statements)

References 49 publications

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“…The stiffness matrix arising from the direct formulation (12) is the transpose to that of the indirect method (11). Theorem 6.44 in [105] gives sufficient conditions for the well-posedness of the direct method. Theorem 7.19 in [21] proves that, for wellposed Dirichlet problems with H 1 (∂ Ω ) data, if the Neumann traces of the trial space coincide with the Dirichlet traces of the test space, then the direct method is wellposed and computes the best approximation of the exact solution in L 2 (∂ Ω ) norm.…”

Section: Single-element Direct and Indirect Trefftz Methodsmentioning

confidence: 99%

A Survey of Trefftz Methods for the Helmholtz Equation

Hiptmair

Moiola

Perugia

2016

Lecture Notes in Computational Science and Engineering

125

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Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces.We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties.One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.

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Section: Single-element Direct and Indirect Trefftz Methodsmentioning

confidence: 99%

A Survey of Trefftz Methods for the Helmholtz Equation

Hiptmair

Moiola

Perugia

2016

Lecture Notes in Computational Science and Engineering

125

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“…(Note that the fact that ·, · Γ is the real duality pairing is crucial; see [54,Lemma 4.10].) By the density of…”

Section: Notation and Basic Resultsmentioning

confidence: 99%

Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

Spence¹

2014

SIAM J. Math. Anal.

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Abstract. We prove wavenumber-explicit bounds on the Dirichlet-to-Neumann map for the Helmholtz equation in the exterior of a bounded obstacle when one of the following three conditions holds: (i) the exterior of the obstacle is smooth and nontrapping, (ii) the obstacle is a nontrapping polygon, or (iii) the obstacle is star-shaped and Lipschitz. We prove bounds on the Neumann-toDirichlet map when condition (i) and (ii) hold. We also prove bounds on the solutions of the interior and exterior impedance problems when the obstacle is a general Lipschitz domain. These bounds are the sharpest yet obtained (for their respective problems) in terms of their dependence on the wavenumber. One motivation for proving these collection of bounds is that they can then be used to prove wavenumber-explicit bounds on the inverses of the standard second-kind integral operators used to solve the exterior Dirichlet, Neumann, and impedance problems for the Helmholtz equation.

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“…The advantage of these two methods, as well as of the method described in this paper, is that they are boundary-based discretizations that do not involve the computation of singular integrals (as opposed to the discretizations of boundary integral equations). Relations between these methods are discussed in detail in [ 32 ], §10 and [ 33 ], §4. In the null-field method, u in ( 2.2 ) is the solution of the Helmholtz equation in the exterior of a bounded obstacle, and v is one of a countable family of separable solutions of the Helmholtz equation in polar coordinates that satisfies the appropriate radiation condition (see, e.g.…”

Section: Discussionmentioning

confidence: 99%

“…(iii) By employing the integral representation and global relations mentioned in (i) and (ii), respectively, it has been possible to obtain exact solutions for a variety of problems for which apparently the usual approaches fail, see e.g. [ 4 , 5 ]. (iv) Ashton [ 6 , 7 ] has developed a rigorous approach for deriving well posedness results for linear elliptic PDEs using the new formalism.…”

Section: Introductionmentioning

confidence: 99%

A numerical technique for linear elliptic partial differential equations in polygonal domains

2015

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Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

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Chapter 6: Overview of Variational Formulations for Linear Elliptic PDEs

Cited by 14 publications

References 49 publications

A Survey of Trefftz Methods for the Helmholtz Equation

A Survey of Trefftz Methods for the Helmholtz Equation

Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

A numerical technique for linear elliptic partial differential equations in polygonal domains

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