Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems |
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Authors: | Pedro Galán del Sastre Rodolfo Bermejo |
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Affiliation: | 1.Departamento de Matemática Aplicada al Urbanismo, a la Edificación y al Medio Ambiente, E.T.S.A.M.,Universidad Politécnica de Madrid,Madrid,Spain;2.Departamento de Matemática Aplicada, E.T.S.I.I.,Universidad Politécnica de Madrid,Madrid,Spain |
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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(Dt2+hm+1(1+\frac\mathopen|logh|Dt))O(\Delta t^{2}+h^{m+1}(1+\frac{\mathopen{|}\log h|}{\Delta t})); and the second, which is O(Dtp+|| (u)\vec]-(u)\vec]h||L¥)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, (u)\vec]\vec{u} is the velocity vector, (u)\vec]h\vec{u}_{h} is the finite element approximation of (u)\vec]\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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