Least‐squares Trefftz‐type elements for the Helmholtz equation

Stojek, Małgorzata

doi:10.1002/(sici)1097-0207(19980315)41:5<831::aid-nme311>3.3.co;2-m

Cited by 10 publications

(16 citation statements)

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“…are T-complete. Other type of T-complete sets for special purposes such as unbounded subdomains and bounded angular corner subdomains can be found in [15].…”

Section: Trefftz Methods: the Ultraweak Variational Formulation And Tmentioning

confidence: 99%

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A comparison of two Trefftz‐type methods: the ultraweak variational formulation and the least‐squares method, for solving shortwave 2‐D Helmholtz problems

Gamallo

Astley

2007

Numerical Meth Engineering

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SUMMARYTrefftz methods for the numerical solution of partial differential equations (PDEs) on a given domain involve trial functions which are defined in subdomains, are generally discontinuous, and are solutions of the governing PDE (or its adjoint) within each subdomain. The boundary conditions and matching conditions between subdomains must be enforced separately. An interesting novel result presented in this paper is that the least-squares method (LSM) and the ultraweak variational formulation, two methods already established for solving the Helmholtz equation, can be derived in the framework of the Trefftz-type methods. In the first case, the boundary conditions and interelement continuity are enforced by means of a least-squares procedure. In the second, a Galerkin-type weighted residual method is used. Another goal of the work is to assess the relative efficiency of each method for solving shortwave problems in acoustics and to study the stability of each method. The numerical performance of each scheme is assessed with reference to two 2-D test problems; acoustic propagation in an uniform soft-walled duct, and propagation in an L-shaped domain, the latter involving singular behaviour at a sharp corner.

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“…are T-complete. Other type of T-complete sets for special purposes such as unbounded subdomains and bounded angular corner subdomains can be found in [15].…”

Section: Trefftz Methods: the Ultraweak Variational Formulation And Tmentioning

confidence: 99%

“…In this case, the discrete problem is obtained from the variational formulation (15). The same discrete subspace K h, p ( ) ⊂ K( ) employed for the discretization of the least-squares formulation is considered…”

Section: The Ultraweak Variational Formulationmentioning

confidence: 99%

A comparison of two Trefftz‐type methods: the ultraweak variational formulation and the least‐squares method, for solving shortwave 2‐D Helmholtz problems

Gamallo

Astley

2007

Numerical Meth Engineering

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“…It is assumed that the scattering form of the Helmholtz equation is to be solved. The theory here follows Stojek [40]. Background theory on suitable functions for the Trefftz method and T-completeness is given by Herrera [41].…”

Section: Trefftz-type Finite Elements For Wavesmentioning

confidence: 99%

Short Waves

Zienkiewicz¹,

Taylor²,

Nithiarasu³

2014

The Finite Element Method for Fluid Dynamics

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“…Compared with the classical finite element method (FEM), the plane wave methods can significantly reduce the required degree of freedom under the same error precision, and with the increase of the wave number, the superiority is more obvious. The weighted plane wave least‐squares method (PWLS) is a frequently used plane wave method . One advantage of PWLS over the other plane wave methods is that the stiffness matrix of the PWLS discrete system is Hermitian positive definite; this lead to solve the resulting system by preconditioned conjugate gradient (PCG) method, and the preconditioner plays an important role in the iterative process.…”

Section: Introductionmentioning

confidence: 99%

“…The weighted plane wave least-squares method (PWLS) is a frequently used plane wave method. [10][11][12][13][14][15] One advantage of PWLS over the other plane wave methods is that the stiffness matrix of the PWLS discrete system is Hermitian positive definite; this lead to solve the resulting system by preconditioned conjugate gradient (PCG) method, and the preconditioner plays an important role in the iterative process.…”

mentioning

confidence: 99%

Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations

Peng

Wang

Shu

2018

Numerical Meth Engineering

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Summary Balancing domain decomposition by constraints algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least‐squares discretization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well‐known discretizations such as the classical finite element method. Through choosing suitable “interface” and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two‐level adaptive balancing domain decomposition by constraints algorithm for the plane wave least‐squares discretization, and the condition number of the preconditioned system is proved to be bounded above by a user‐defined tolerance and a constant that is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large‐scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.

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Least‐squares Trefftz‐type elements for the Helmholtz equation

Cited by 10 publications

References 0 publications

A comparison of two Trefftz‐type methods: the ultraweak variational formulation and the least‐squares method, for solving shortwave 2‐D Helmholtz problems

A comparison of two Trefftz‐type methods: the ultraweak variational formulation and the least‐squares method, for solving shortwave 2‐D Helmholtz problems

Short Waves

Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations

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