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 have been proved to lead to high order numerical methods, and the quasi-Trefftz property is critical for the numerical analysisof these methods . The present work introduces a new family of GPWs  . The motivation to introduce these new GPWs, amplitude-based, lies in the poor behavior of the phase-based GPWs in the pre-asymptotic regime, which will be addressed by avoiding high degree polynomials within an exponential. The new ansatz is still based on a plane wave function, but it introduces higher order terms in the amplitude rather than the phase of the plane wave as was initially proposed. The new functions' construction and  study of their interpolation properties follow the roadmap proposed in a previous work . For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially varying wavenumber, and follows the extension to a range of operators allowing for anisotropy in the first and second order terms  . Numerical simulations illustrate the theoretical study of the new quasi-Trefftz functions.
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 pre-asymptotic regime, which will be tamed by avoiding high degree polynomials within an exponential. The new ansatz is introduces higher order terms in the amplitude rather than the phase of a plane wave as was initially proposed. The new functions' construction and the study of their interpolation properties are guided by the roadmap proposed in 16 . For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially-varying wavenumber. 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.
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