Johnny Guzmán

We construct a local Fortin projection for the Scott–Vogelius finite element pair for polynomial degree k ≥ 4 on general shape-regular triangulations in two dimensions. In particular, the triangulation may contain singular vertices. In addition to preserving the divergence in the dual of the pressure space, the projection preserves discrete boundary data and satisfies local stability estimates.

Mixed methods for linear elasticity with strongly symmetric stresses of lowest order are studied in this paper. On each simplex, the stress space has piecewise linear components with respect to its Alfeld split (which connects the vertices to barycenter), generalizing the Johnson–Mercier two-dimensional element to higher dimensions. Further reductions in the stress space in the three-dimensional case (to 24 degrees of freedom per tetrahedron) are possible when the displacement space is reduced to local rigid displacements. Proofs of optimal error estimates of numerical solutions and improved error estimates via postprocessing and the duality argument are presented.

We consider a loosely coupled, non-iterative Robin-Robin coupling method proposed and analyzed in [Numer. Algorithms, 99:921-948, 2025] for a parabolic-parabolic interface problem. We modify the first step of the scheme so that several error difference quantities remain higher order convergence without requiring additional assumptions. Numerical results are presented to support our findings.

We construct projections onto the classical finite element spaces based on Lagrange, Nédélec, Raviart-Thomas, and discontinuous elements on shape-regular simplicial meshes. Our projections are defined locally, are bounded in the L2-norm, and commute with the corresponding differential operators. The cornerstone of the construction are local weight functions which are piecewise polynomials built using the Alfeld split of local patches from the original simplicial mesh. This way, the L2-stability of the projections is established by invoking discrete Poincaré inequalities on these local stars, for which we provide constructive proofs. We also show how to modify the construction to preserve homogeneous boundary conditions. The material is presented using the language of vector calculus, and links to the formalism of finite element exterior calculus are provided.

We investigate discrete Poincaré inequalities on piecewise polynomial subspaces of the Sobolev spaces H(curl, ω) and H(div, ω) in three space dimensions. We characterize the dependence of the constants on the continuous-level constants, the shape regularity and cardinality of the underlying tetrahedral mesh, and the polynomial degree. One important focus is on meshes being local patches (stars) of tetrahedra from a larger tetrahedral mesh. We also review various equivalent results to the discrete Poincaré inequalities, namely stability of discrete constrained minimization problems, discrete inf-sup conditions, bounds on operator norms of piecewise polynomial vector potential operators (Poincaré maps), and existence of graph-stable commuting projections.

In this paper, we continue the work on implicit–explicit (IMEX) time discretizations for the incompressible Oseen’s equations that we started in Burman et al. [Implicit–explicit time discretization for Oseen’s equation at high Reynolds number with application to fractional step methods, SIAM J. Numer. Anal. 61 (2023) 2859–2886]. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein, we focus on the IMEX Crank–Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensure inf–sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions, we prove the stability of the IMEX Crank–Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order [ ]. Here [ ] is the mesh parameter, [ ] the polynomial order and [ ] the time step. Finally, we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier–Stokes’ equations.