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1. Superconvergent and Divergence-Free Finite Element Methods for the Stokes Equation

   Chen, Long; Huang, Xuehai; Zhang, Chao; Zhao, Xinyue (May 2026, SIAM journal on numerical analysis)

   This paper develops divergence-free mixed finite element methods for the Stokes equation. Using H(div)-conforming velocities and discontinuous pressures ensures the inf-sup condition for the velocity--pressure pair and yields pointwise divergence-free velocities. However, this choice makes the vector Laplacian difficult to discretize. Inspired by mass-conserving mixed formulations with stresses, tangential--normal continuous traceless tensor elements are introduced to discretize the vector Laplacian. An inf-sup condition for the weak div operator between the stress and velocity spaces is then proved. Two key properties characterize the scheme. First, the stress--velocity inf-sup stability gives a stable discretization of the vector Laplacian without additional stabilization, unlike discontinuous Galerkin or virtual element methods. Second, the scheme has the property that if a stress field is distributionally divergence-free against the discrete divergence-free velocity space, then it is also distributionally divergence-free against the continuous divergence-free velocity space. This property decouples the stress and velocity errors and leads to superconvergence. As a result, optimal-order error estimates are obtained for the stress, while the velocity and pressure converge at rates higher than the approximation orders of the chosen spaces. Numerical experiments confirm the theoretical results. 

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2. Robust error analysis of stabilized linear EMAC-ESAV finite element schemes for the incompressible Navier-Stokes equations

   https://doi.org/10.1090/mcom/4087  

   Lan, Rihui; Liu, Mengyao; Ju, Lili (May 2025, Mathematics of Computation)

   In this paper, we propose and study first- and second-order (in time) stabilized linear finite element schemes for the incompressible Navier-Stokes (NS) equations. The energy, momentum, and angular momentum conserving (EMAC) formulation has emerged as a promising approach for conserving energy, momentum, and angular momentum of the NS equations, while the exponential scalar auxiliary variable (ESAV) has become a popular technique for designing linear energy-stable numerical schemes. Our method leverages the EMAC formulation and the Taylor-Hood element with grad-div stabilization for spatial discretization. We adopt the implicit-explicit backward differential formulas (BDFs) coupled with a novel stabilized ESAV approach for time stepping. For the solution process, we develop an efficient decoupling technique for the resulting fully-discrete systems so that only one linear Stokes solve is needed at each time step, which is similar to the cost of classic implicit-explicit BDF schemes for the NS equations. Robust optimal error estimates are successfully derived for both velocity and pressure for the two proposed schemes, with Gronwall constants that are particularly independent of the viscosity. Furthermore, it is rigorously shown that the grad-div stabilization term can greatly alleviate the viscosity-dependence of the mesh size constraint, which is required for error estimation when such a term is not present in the schemes. Various numerical experiments are conducted to verify the theoretical results and demonstrate the effectiveness and efficiency of the grad-div and ESAV stabilization strategies and their combination in the proposed numerical schemes, especially for problems with high Reynolds numbers. 

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3. An H(div)-conforming Finite Element Method for Biot’s Consolidation Model

   Zeng, Yuping; Cai, Mingchao; Wang, Feng. (August 2019, East Asian Journal on Applied Mathematics)

   An H(div)-conforming finite element method for the Biot’s consolidation mo- del is developed, with displacements and fluid velocity approximated by elements from BDM_k space. The use of H(div)-conforming elements for flow variables ensures the local mass conservation. In the H(div)-conforming approximation of displacement, the tan- gential components are discretised in the interior penalty discontinuous Galerkin frame- work,and the normal components across the element interfaces are continuous. Having introduced a spatial discretisation, we develop a semi-discrete scheme and a fully dis- crete scheme,prove their unique solvability and establish optimal error estimates for each variable. 

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4. Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in H (div)

   https://doi.org/10.1051/m2an/2025045  

   Pazner, Will (July 2025, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, we construct and analyze preconditioners for the interior penalty discontinuous Galerkin discretization posed in the spaceH(div). These discretizations are used as one component in exactly divergence-free pressure-robust discretizations for the Stokes problem. Three preconditioners are presently considered: a subspace correction preconditioner using vertex patches and the lowest-orderH¹-conforming space as a coarse space, a fictitious space preconditioner using the degree-pdiscontinuous Galerkin space, and an auxiliary space preconditioner using the degree-(p− 1) discontinuous Galerkin space and a block Jacobi smoother. On certain classes of meshes, the subspace and fictitious space preconditioners result in provably well-conditioned systems, independent of the mesh sizeh, polynomial degreep, and penalty parameterη. All three preconditioners are shown to be robust with respect tohon general meshes, and numerical results indicate that the iteration counts grow only mildly with respect topin the general case. Numerical examples illustrate the convergence properties of the preconditioners applied to structured and unstructured meshes. These solvers are used to construct block-diagonal preconditioners for the Stokes problem, which result in uniform convergence when used with MINRES. 

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5. Matrix-Free Stabilized BDF Schemes for Semilinear Parabolic Equations with Unconditional Maximum Bound Principle Preservation and Energy Stability

   Dai, Haishen; Lei, Huan; Zheng, Bin (March 2026, ArXivorg)

   We develop a family of stabilized backward differentiation formula (sBDF) schemes of orders one through four for semilinear parabolic equations. The proposed methods are designed to achieve three properties that are rarely available simultaneously in high-order time discretizations: unconditional preservation of the maximum bound principle (MBP), unconditional discrete energy stability, and practical matrix-free implementation. The construction integrates carefully designed stabilization terms, fixed-point iterations, and a pointwise cut-off strategy. The nonlinear algebraic systems arising from the implicit sBDF discretizations are solved by fixed-point iteration, resulting in fully matrix-free algorithms. This makes the approach particularly attractive for practical computations on general domains and under mixed boundary conditions, where FFT-based exponential time differencing methods are often unavailable or inefficient. We further present a unified analysis for the fully implemented schemes, explicitly incorporating the interplay among time discretization, nonlinear iteration, and cutoff. Unconditional contractivity of the fixed-point iterations and error estimates are established. For the Allen–Cahn equation, we additionally prove an unconditional discrete energy dissipation law. Numerical experiments confirm the theoretical convergence rates and demonstrate the robustness and efficiency of the proposed methods, particularly relative to ETD-based approaches for problems with mixed boundary conditions. 

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