Hajduk HKuzmin DLube GÖffner P2025-05-012025-05-012025-05-30Hajduk H, Kuzmin D, Lube G, Öffner P. (2025). Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations. Computers and Fluids. 294.0045-7930https://mro.massey.ac.nz/handle/10179/72832We show that finite element discretizations of incompressible flow problems can be designed to ensure preservation/dissipation of kinetic energy not only globally but also locally. In the context of equal-order (piecewise-linear) interpolations, we prove the validity of a semi-discrete energy inequality for a quadrature-based approximation to the nonlinear convective term, which we combine with the Becker–Hansbo pressure stabilization. An analogy with entropy-stable algebraic flux correction schemes for the compressible Euler equations and the shallow water equations yields a weak ‘bounded variation’ estimate from which we deduce the semi-discrete Lax–Wendroff consistency and convergence towards dissipative weak solutions. The results of our numerical experiments for standard test problems confirm that the method under investigation is non-oscillatory and exhibits optimal convergence behavior.(c) 2025 The Author/sCC BY 4.0https://creativecommons.org/licenses/by/4.0/Incompressible Euler and Navier–Stokes equationsStabilized finite element methodsEqual-order interpolationEnergy inequalityConsistencyConvergenceDissipative weak solutionsJournal article10.1016/j.compfluid.2025.1066221879-0747journal-article106622S0045793025000829