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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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HOMOGENIZATION OF QUASI-CRYSTALLINE FUNCTIONALS VIA TWO-SCALE-CUT-AND-PROJECT CONVERGENCE
We consider a homogenization problem associated with quasi-crystalline multiple inte- grals of the form uε ∈ Lp(Ω;Rd) 7→ ˆΩ fR x, xε,uε(x) dx, where uε is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, fR, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields uε that are in the kernel of a given constant-coefficient linear partial differential operator, A, that is, Auε = 0. Our results provide a generalization of related ones in the literature concerning the A = curl case to more general differential operators A with constant coefficients, and without coercivity assumptions on the Lagrangian fR.
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- PAR ID:
- 10346292
- Date Published:
- Journal Name:
- SIAM journal on mathematical analysis
- Volume:
- 53
- ISSN:
- 1095-7154
- Page Range / eLocation ID:
- 1785 - 1817
- 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/10.1137/20M1341222
