Error Estimates for a Combined Finite Volume-Finite Element Method for Nonlinear Convection-Diffusion Problems

• Published in: SIAM journal on numerical analysis Vol. 36; no. 5; pp. 1528 - 1548
• Main Authors: Feistauer, Miloslav, Felcman, Jiri, Maria Lukacova-Medvid'Ova, Warnecke, Gerald
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 1999
• Subjects: Approximation; Boundary value problems; Computational methods in fluid dynamics; Conservation laws; Error rates; Estimates; Estimation methods; Exact sciences and technology; Finite element method; Fluid dynamics; Fundamental areas of phenomenology (including applications); Heat conductivity; Investigations; Mathematical analysis; Mathematics; Maximum principle; Methods; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Physics; Sciences and techniques of general use; Triangulation; Vertices; Viscosity; Solution uniqueness; Neumann problem; Numerical integration; Error estimation; Existence of solution; Finite volume method; Maximum principle; Convection diffusion equation; Partial differential equation; Non linear equation; Finite element method; Boundary value problem; Discretization; Navier Stokes equation; Lagrange interpolation; A priori estimation; Dirichlet problem; Mixed problem; Aubin Nitsche duality method; Lax Friedrichs scheme; Engquist Osher scheme; Green theorem; Compressible flow
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142997314695

The subject of this paper is the analysis of error estimates of the combined finite volume-finite element (FV-FE) method for the numerical solution of a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid, whereas the diffusion term is discretized by piecewise linear conforming triangular finite elements. Under the assumption that the exact solution possesses some regularity properties and the triangulations are of a weakly acute type, with the aid of the discrete maximum principle and a priori estimates, error estimates of the method are proved.