A Spectral Mimetic Least-Squares Method for Generalized Convection-Diffusion Problems
We present a spectral mimetic least-squares method for a model convection-diffusion problem, which preserves conservation properties. The problem is solved using differential geometry where the topological part and the constitutive part have been separated. It is shown that the topological part is solved exactly independent of the order of the spectral expansion. The mimetic method incorporates the Lie derivative for the convective term, by means of Cartans homotopy formula, see for example Abraham et al. (1988) (Manifolds, Tensor Analysis, and Applications, Springer, New York). The spectral mimetic least-squares method is compared to a more classic spectral least-squares method. It is shown that both schemes lead to spectral convergence.
- 3.P. Bochev, J.M. Hyman, Principles of mimetic discretizations of differential operators, in Compatible Spatial Discretizations, ed. by D.N. Arnold et al. (Springer, New York, 2006)Google Scholar
- 5.C. Canuto et al., Spectral Methods: Fundamentals in Single Domains, 1st edn. Scientific Computation (Springer, Berlin, 2007)Google Scholar
- 6.M. Desbrun et al. Discrete exterior calculus. ArXiv Mathematics e-prints (2005)Google Scholar
- 7.M. Gerritsma, Edge functions for spectral element methods, in Spectral and High Order Methods for Partial Differential Equations: Selected Papers from the ICOSAHOM ’09 Conference, June 22–26, Trondheim, Norway, ed. by S. Jan Hesthaven, M. Einar Rønquist (Springer, Berlin, 2011), pp. 199–207CrossRefGoogle Scholar
- 11.A. McInerney, First Steps in Differential Geometry: Riemannian, Contact, Symplectic, 1st edn. Undergraduate Texts in Mathematics (Springer, New York, 2013)Google Scholar