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1. $$L^2(I;H^1(\Omega)^d)$$ and $$L^2(I;L^2(\Omega)^d)$$ best approximation type error estimates for Galerkin solutions of transient Stokes problems

   Leykekhman, Dmitriy; Vexler, Boris (October 2023, Calcolo)

   In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in $$L^2(I;L^2(\Omega)^d)$$ and $$L^2(I;H^1(\Omega)^d)$$ norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type $$L^2(I;H^1(\Omega))$$ error estimate are new even for scalar parabolic problems. 

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2. Stokes flow of an evolving fluid film with arbitrary shape and topology

   https://doi.org/10.1017/jfm.2024.1208  

   Zhu, Cuncheng; Saintillan, David; Chern, Albert (January 2025, Journal of Fluid Mechanics)

   The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modelling of lipid bilayers in cells. While the governing equations were formulated by Scriven (1960), solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and translates directly to discretization. We construct a discrete analogue of the system using Onsager's variational principle, which, in a smooth context, governs the flow of a viscous medium. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds. 

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3. A cutFEM divergence–free discretization for the stokes problem

   https://doi.org/10.1051/m2an/2022072  

   Liu, Haoran; Neilan, Michael; Olshanskii, Maxim (January 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O ( h ) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O ( h −1 ), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates. 

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4. On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case

   https://doi.org/10.1142/S0218202520500438  

   Li, Xiaoli; Shen, Jie (November 2020, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme. 

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5. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness

   https://doi.org/10.1007/s10884-021-10013-5  

   Hupkes, H. J.; Van Vleck, E. S. (June 2022, Journal of Dynamics and Differential Equations)

   Abstract In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations. 

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