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

  • Authors:

  • Pedro Galán Del Sastre;Rodolfo Bermejo

  • Affiliations:

  • Departamento de Matemática Aplicada al Urbanismo, a la Edificación y al Medio Ambiente, E.T.S.A.M., Universidad Politécnica de Madrid, Madrid, Spain 28040;Departamento de Matemática Aplicada, E.T.S.I.I., Universidad Politécnica de Madrid, Madrid, Spain 28006

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2011

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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 驴2. 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.