TY - JOUR
T1 - Amplitude-based generalized plane waves
T2 - New quasi-trefftz functions for scalar equations in two dimensions
AU - Imbert-Gerard, Lise Marie
N1 - Funding Information:
∗Received by the editors September 18, 2020; accepted for publication (in revised form) March 10, 2021; published electronically June 16, 2021. https://doi.org/10.1137/20M136791X Funding: The work of the author was supported by National Science Foundation grant DMS-1818747. †University of Arizona, Tucson, AZ 85721 USA ([email protected], https://www.math. arizona.edu/∼lmig/).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics
PY - 2021
Y1 - 2021
N2 - Generalized plane waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz property, i.e., they are not exact solutions to the governing equation, they instead satisfy a quasi-Trefftz property: They are only approximate solutions. They lead to high-order numerical methods, and this quasi-Trefftz property is critical for their numerical analysis. The present work introduces a new family of GPWs: amplitude-based. The motivation lies in the poor behavior of the phase-based GPW approximations in the preasymptotic regime, which will be tamed by avoiding high-degree polynomials within an exponential. The new ansatz introduces higher-order terms in the amplitude rather than in the phase of a plane wave as was initially proposed. The new functions' construction and the study of their approximation properties are guided by the road map proposed in [L.-M. Imbert-Gérard and G. Sylvand, Numer. Math., to appear]. For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially varying wave number. The extension to a range of operators allowing for anisotropy in the first- and second-order terms follows. Numerical simulations illustrate the theoretical study of the new quasi-Trefftz functions.
AB - Generalized plane waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz property, i.e., they are not exact solutions to the governing equation, they instead satisfy a quasi-Trefftz property: They are only approximate solutions. They lead to high-order numerical methods, and this quasi-Trefftz property is critical for their numerical analysis. The present work introduces a new family of GPWs: amplitude-based. The motivation lies in the poor behavior of the phase-based GPW approximations in the preasymptotic regime, which will be tamed by avoiding high-degree polynomials within an exponential. The new ansatz introduces higher-order terms in the amplitude rather than in the phase of a plane wave as was initially proposed. The new functions' construction and the study of their approximation properties are guided by the road map proposed in [L.-M. Imbert-Gérard and G. Sylvand, Numer. Math., to appear]. For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially varying wave number. The extension to a range of operators allowing for anisotropy in the first- and second-order terms follows. Numerical simulations illustrate the theoretical study of the new quasi-Trefftz functions.
KW - Best approximation properties
KW - Generalized plane waves
KW - Quasi-Trefftz methods
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U2 - 10.1137/20M136791X
DO - 10.1137/20M136791X
M3 - Article
AN - SCOPUS:85108667408
SN - 0036-1429
VL - 59
SP - 1663
EP - 1686
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 3
ER -