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Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.
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Parameterization of Unstable Manifolds for DDEs: Formal Series Solutions and Validated Error Bounds
This paper studies the local unstable manifold attached to an equilibrium solution of a system of delay differential equations (DDEs). Two main results are developed. The first is a general method for computing the formal Taylor series coefficients of a function parameterizing the unstable manifold. We derive linear systems of equations whose solutions are the Taylor coefficients, describe explicit formulas for assembling the linear equations for DDEs with polynomial nonlinearities. We also discuss a scheme for transforming non-polynomial DDEs into polynomial ones by appending auxiliary equations. The second main result is an a-posteriori theorem which—when combined with deliberate control of rounding errors— leads to mathematically rigorous computer assisted convergence results and error bounds for the truncated series. Our approach is based on the parameterization method for invariant manifolds and requires some mild non-resonance conditions between the unstable eigenvalues
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- Award ID(s):
- 1813501
- PAR ID:
- 10274131
- Date Published:
- Journal Name:
- Journal of Dynamics and Differential Equations
- ISSN:
- 1040-7294
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10884-021-10002-8
