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Abstract We consider a parabolic–parabolic interface problem and construct a loosely coupled prediction-correction scheme based on the Robin–Robin splitting method analyzed in [J. Numer. Math., 31(1):59–77, 2023]. We show that the errors of the correction step converge at $$\mathcal O((\varDelta t)^{2})$$, under suitable convergence rate assumptions on the discrete time derivative of the prediction step, where $$\varDelta t$$ stands for the time-step length. Numerical results are shown to support our analysis and the assumptions. 

We construct conforming finite element elasticity complexes on Worsey–Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey–Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space. 

We hybridize the methods of finite element exterior calculus for the Hodge-Laplace problem on differential k-forms in ℝn. In the cases k=0 and k=n, we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for 0 

We construct several smooth finite element spaces defined on three-dimensional Worsey–Farin splits. In particular, we construct C 1 C^1 , H 1 ( curl ) H^1(\operatorname {curl}) , and H 1 H^1 -conforming finite element spaces and show the discrete spaces satisfy local exactness properties. A feature of the spaces is their low polynomial degree and lack of extrinsic supersmoothness at subsimplices of the mesh. In the lowest order case, the last two spaces in the sequence consist of piecewise linear and piecewise constant spaces, and are suitable for the discretization of the (Navier-)Stokes equation. 

Abstract We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two ‘almost’ straight lines.