An hp finite element method for convection-diffusion problems in one dimension

Abstract

We analyse an hp FEM for convection-diffusion problems. Stability is achieved by suitably upwinded test functions, generalizing the classical α-quadratically upwinded and the Hemker test-functions for piecewise linear trial spaces (see, e.g., Morton 1995 Numerical Solutions of Convection-Diffusion Problems, Oxford: Oxford University Press, and the references therein). The method is proved to be stable independently of the viscosity. Further, the stability is shown to depend only weakly on the spectral order. We show how suf"ciently accurate, approximate upwinded test functions can be computed on each element by a local least-squares FEM. Under the assumption of analyticity of the input data, we prove robust exponential convergence of the method. Numerical experiments con"rm our convergence estimates and show robust exponential convergence of the hp FEM even for viscosities of the order of the machine precision, i.e., for the limiting transport problem.

Optimal a priori error estimates for the $hp$-version of the local discontinuous Galerkin method for convection--diffusion problems

“…A remedy is to employ a needle-element of the appropriate width or geometric mesh refinement near the boundary. It has been shown recently by Melenk [16], Melenk and Schwab [17,19], Schwab and Suri [25] and Wihler and Schwab [32] that the use of these mesh-design principles yields exponential rates of convergence that are robust with respect to the diffusivity parameter d. We demonstrate this robustness in our numerical examples in section 5.5 below. [5].…”

On the robust exponential convergence of hp finite element methods for problems with boundary layers

“…Of course, as the solutions of the convection-diffusion equation have a boundary layer at the outflow boundary only, it would be enough to use two elements where the small element is located at the outflow boundary. Let us conclude our remarks on the convection-diffusion equation by stressing that stability of finite element methods for the convection-diffusion equation is, as opposed to the reaction-diffusion equation considered in this paper, a non-trivial issue; a stable hp FEM for the convection-diffusion equation able to make use of robust exponential approximability results of the type proved in this paper is presented by Melenk & Schwab (1997b).…”

“…Whereas the approximability results of this paper hold true for the convectiondiffusion equation as well, the stability of finite element methods for that equation is a non-trivial issue. A stable hp method for the convection-diffusion equation featuring robust exponential rates of convergence is presented by Melenk & Schwab (1997b).…”

Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

“…It would be desirable to develop error estimates uniform with respect to ν. However, this has been obtained only in very few works analyzing simple problems under rather special assumptions when complete analytic behaviour of solutions is known ( [1], [35] and citations in [39]). …”