Boundary methods for solving elliptic problems with singularities and interfaces
SIAM Journal on Numerical Analysis
The algebraic eigenvalue problem
The algebraic eigenvalue problem
Boundary approximation methods for solving elliptic problems on unbounded domains
Journal of Computational Physics
A least-squares finite element method for the Helmholtz equation
Computer Methods in Applied Mechanics and Engineering
Finite element methods for the Helmholtz equation in an exterior domain: model problems
Computer Methods in Applied Mechanics and Engineering
The effective condition number applied to error analysis of certain boundary collocation methods
Journal of Computational and Applied Mathematics
SIAM Journal on Numerical Analysis
A domain decomposition method for the Helmholtz equation and related optimal control problems
Journal of Computational Physics
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
A Fast Spectral Solver for a 3D Helmholtz Equation
SIAM Journal on Scientific Computing
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Multigrid Simulation for High-Frequency Solutions of the Helmholtz Problem in Heterogeneous Media
SIAM Journal on Scientific Computing
Journal of Computational Physics
Spectral Approximation of the Helmholtz Equation with High Wave Numbers
SIAM Journal on Numerical Analysis
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Error analysis of the Trefftz method for solving Laplace's eigenvalue problems
Journal of Computational and Applied Mathematics
On solution uniqueness of elliptic boundary value problems
Journal of Computational and Applied Mathematics
The method of external excitation for solving generalized Sturm-Liouville problems
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Numerical implementation of the EDEM for modified Helmholtz BVPs on annular domains
Journal of Computational and Applied Mathematics
Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues
SIAM Journal on Numerical Analysis
Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation
SIAM Journal on Numerical Analysis
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The Trefftz method (TM) [E. Trefftz, Ein Gegenstuck zum Ritz'schen Verfahren, in: Proc. 2nd Ind. Congr. Appl. Mech., Zurich, 1926, pp. 131-137] (i.e., Boundary approximation method) is developed to solve the Helmholtz equation, @Du+k^2u=0, where k^2 is not exactly equal (but may be very close) to an eigenvalue of the operator -@D. Piecewise particular solutions are chosen and then matched together in order to satisfy the exterior and interior boundary conditions. Error analysis is presented to estimate error bounds for the Helmholtz solutions in the entire solution domain. Let @d be the smallest relative distance between k^2 and the eigenvalues of -@D. We prove that the error asymptote of the solutions by the TM is O(1@d) as @d-0, which is called degeneracy in this paper. Such an asymptote O(1@d) has been verified by the numerical computations. We also explain why the exponential convergence rates of solutions can be obtained easily by splitting the solution domain into smaller subdomains. Numerical experiments of both smooth and singular solutions are provided, to support the TM algorithms and the error analysis made.