  We describe and analyze a quasi-Trefftz DG method for solving boundary value
problems for the homogeneous diffusion-advection-reaction equation with
piecewise-smooth coefficients. Trefftz schemes are high-order Galerkin methods
whose discrete functions are elementwise exact solutions of the underlying PDE.
Trefftz basis functions can be computed for many PDEs that are linear,
homogeneous and with piecewise-constant coefficients. However, if the equation
has varying coefficients, in general, exact solutions are unavailable, hence
the construction of discrete Trefftz spaces is impossible. Quasi-Trefftz
methods have been introduced to overcome this limitation, relying on discrete
spaces of functions that are elementwise "approximate solutions" of the PDE. A
space-time quasi-Trefftz DG method for the acoustic wave equation with smoothly
varying coefficients has recently been studied; since it has shown excellent
results, we propose a related method that can be applied to second-order
elliptic equations. The DG weak formulation is derived using an interior
penalty parameter and the upwind numerical fluxes. We choose polynomial
quasi-Trefftz basis functions, whose coefficients can be computed with a simple
algorithm based on the Taylor expansion of the PDE's coefficients. The main
advantage of Trefftz and quasi-Trefftz schemes over more classical ones is the
higher accuracy for comparable numbers of degrees of freedom. We prove that the
dimension of the quasi-Trefftz space is smaller than the dimension of the full
polynomial space of the same degree and that yields the same optimal
convergence rates. The quasi-Trefftz DG method is well-posed, consistent and
stable and we prove its high-order convergence. We present some numerical
experiments in two dimensions that show excellent properties in terms of
approximation and convergence rate.
