A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

Abstract

We consider a time‐dependent and a steady linear convection‐diffusion‐reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin–Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element–finite volume method: the diffusion term is discretized by Crouzeix–Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. The L 2 ( H 1 ) ‐ and the L ∞ ( L 2 ) ‐error in the unsteady case and the H1‐error in the steady one are estimated against the data, in such a way that no parameter enters exponentially into the constants involved. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1591–1621, 2016

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

“…Thus our scheme does allow to maintain accuracy. Reference [12] generalizes earlier results from [13], where the case b constant, m = 0, Γ D = ∂Ω, f D = 0, a = −ν (δ jk ) 1≤j,k≤2 is considered.…”

“…Of course, one might ask whether this term degrades accuracy. These doubts, however, are lifted by a companion paper [12], where we show that optimal error estimates hold with respect to the L 2 (H 1 )-and the L ∞ (L 2 )-norm (in the steady case: with respect to the H 1 -norm). Thus our scheme does allow to maintain accuracy.…”

“…Alternatively, this theorem may be deduced from ( [11], Thm. 3.1.4); see ( [12], proof of Thm. 2.2).…”

“…Well known: by using a scaling argument as in the proof of Lemma 2.7, we may reduce Lemma 2.8 to a trace estimate on the reference triangleK; see ( [12], proof of Lem. 2.1) for details.…”