A Priori Error Analysis of Residual-Free Bubbles for Advection-Diffusion Problems

• Published in: SIAM journal on numerical analysis Vol. 36; no. 6; pp. 1933 - 1948
• Main Authors: Brezzi, F., Hughes, T. J. R., Marini, L. D., Russo, A., Suli, E.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 1999
• Subjects: Accuracy; Analytical estimating; Applied mathematics; Approximation; Bubbles; Computational methods in fluid dynamics; Convection and heat transfer; Economic bubbles; Error analysis; Error bounds; Estimation methods; Exact sciences and technology; Finite element method; Fluid dynamics; Fundamental areas of phenomenology (including applications); Inequality; Mathematical functions; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, boundary value problems; Physics; Sciences and techniques of general use; Turbulent flows, convection, and heat transfer; Vertices; Finite element method; Lax Milgram lemma; Cauchy Schwarz inequality; Error estimation; Stabilization; A priori estimation; Maximum principle; Dirichlet problem; Advection diffusion equation; Galerkin method; Algorithm; Implementation
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142998342367

We develop an a priori error analysis of a finite element approximation to the elliptic advection-diffusion equation -εΔ u + α· ∇ u = f subject to a homogeneous Dirichlet boundary condition, based on the use of residual-free bubble functions. An optimal order error bound is derived in the so-called stability-norm (ε|∇ v|2 L2(Ω)+ ∑ hTh T|a·∇ v|2 L2(T))1/2, where hTdenotes the diameter of element T in the subdivision of the computational domain.