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Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems

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Abstract

We present in this paper an analysis of a semi-Lagrangian second order Backward Difference Formula combined with hp-finite element method to calculate the numerical solution of convection diffusion equations in ℝ2. Using mesh dependent norms, we prove that the a priori error estimate has two components: one corresponds to the approximation of the exact solution along the characteristic curves, which is \(O(\Delta t^{2}+h^{m+1}(1+\frac{\mathopen{|}\log h|}{\Delta t}))\); and the second, which is \(O(\Delta t^{p}+\| \vec{u}-\vec{u}_{h}\|_{L^{\infty}})\), represents the error committed in the calculation of the characteristic curves. Here, m is the degree of the polynomials in the finite element space, \(\vec{u}\) is the velocity vector, \(\vec{u}_{h}\) is the finite element approximation of \(\vec{u}\) and p denotes the order of the method employed to calculate the characteristics curves. Numerical examples support the validity of our estimates.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada al Urbanismo, a la Edificación y al Medio Ambiente, E.T.S.A.M., Universidad Politécnica de Madrid, Avda. Juan de Herrera 4, 28040, Madrid, Spain

    Pedro Galán del Sastre

  2. Departamento de Matemática Aplicada, E.T.S.I.I., Universidad Politécnica de Madrid, C/ José Gutiérrez Abascal 2, 28006, Madrid, Spain

    Rodolfo Bermejo

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  1. Pedro Galán del Sastre
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Correspondence to Pedro Galán del Sastre.

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Galán del Sastre, P., Bermejo, R. Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems. J Sci Comput 49, 211–237 (2011). https://doi.org/10.1007/s10915-010-9454-2

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