Deriving Locally Conservative Fluxes From High‐Order Finite Element Solutions by Solving a Constrained Minimization Problem

• Published in: Numerical methods for partial differential equations Vol. 41; no. 5
• Main Authors: Gong, Wenbo, Cao, Waixiang, Zou, Qingsong
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.09.2025; Wiley Subscription Services, Inc
• Subjects: Conservation laws; conservative flux; Constraints; Finite element method; finite element methods; finite volume methods; Fluxes; Linear algebra; Mathematical analysis; minimization; Optimization; Polynomials; Porous media; Sobolev space
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.70035

ABSTRACT In this article, the classical kth$$ k\mathrm{th} $$ order, k≥1$$ k\ge 1 $$, Galerkin finite element solution for convection‐diffusion equations is post‐processed to derive the numerical fluxes which are conservative on a prescribed set of control volumes. The post‐processing technique is realized through solving a constrained minimization problem where the constraints arise naturally from local conservation laws taking place on the control volumes, and the numerical flux is taken to be a polynomial of order k−1$$ k-1 $$ defined at each edge of control volumes. In contrast to the classic mixed method, the linear algebraic system derived from our minimization problem is symmetric. It is shown that the numerical flux converges to the exact one with optimal orders in a certain Sobolev spaces. In addition, we correct the finite element solution so that the constitute law is satisfied in some weak sense. Numerical experiments are conducted to confirm our theoretic findings. In particular, a simplified two‐phase flow in highly heterogeneous porous media is simulated with our developed numerical flux.