The finite volume-complete flux scheme for one-dimensional advection-diffusion-reaction equations

We present a new integral representation for the flux of the advection-diffusion-reaction equation, which is based on the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme, which is second order accurate, uniformly in the local Peclet numbers. The flux approximation is combined with a finite volume method, and the resulting finite volume-complete flux scheme is validated for several test problems

The complete flux scheme in cylindrical coordinates

We consider the complete ¿ux (CF) scheme, a ¿nite volume method (FVM) presented in [1]. CF is based on an integral representation for the ¿uxes, found by solving a local boundary value problem that includes the source term. It performs well (second order accuracy) for both diffusion and advection dominated problems. In this paper we focus on cylindrically symmetric conservation laws of advection-diffusion-reaction type. [1] ten Thije Boonkkamp, J.H.M., Anthonissen, M.J.H.: The ¿nite volume-complete ¿ux scheme for advection-diffusion-reaction equations. Journal of Scienti¿c Computing 46(1), 47–70 (2011

A posteriori error bounds for discontinuous Galerkin methods for quasilinear parabolic problems

We derive a posteriori error bounds for a quasilinear parabolic problem, which is approximated by the $hp$-version interior penalty discontinuous Galerkin method (IPDG). The error is measured in the energy norm. The theory is developed for the semidiscrete case for simplicity, allowing to focus on the challenges of a posteriori error control of IPDG space-discretizations of strictly monotone quasilinear parabolic problems. The a posteriori bounds are derived using the elliptic reconstruction framework, utilizing available a posteriori error bounds for the corresponding steady-state elliptic problem.Comment: 8 pages, conference ENUMATH 200

Fresnel reflections in inverse double freeform lens design

In this paper we present a method for designing a double freeformlens that includes the effect of Fresnel reflections on the output intensity.We elaborate this method for the case of a point source and a far-field target. A new expression for the transmittance through a double freeform lens is derived, and we adapt a least-squares algorithm to account for this transmittance. A test case based on street lighting is used to show that our adaptation improves the accuracy of the algorithm and that it is possible to minimize Fresnel losses with this new method to design efficient lenses.</p