Abstract Sequences of block matrices with rectangular Toeplitz blocks arise in several applications, including the numerical discretization of differential problems. We compute the (asymptotic) spectral and singular value distribution of these sequences in the case where the asymptotic block size ratios are not necessarily rational as in [A. Adriani, I. Furci, C. Garoni, and S. Serra-Capizzano, Spectral and singular value distribution of sequences of block matrices with rectangular Toeplitz blocks. Part I: Asymptotically rational block size ratios. J. Numer. Math. DOI: 10.1515/jnma-2025-0091]. Our derivation relies on the recent notion of generalized approximating classes of sequences as well as on a functional-based approach to describe the spectral and singular value distributions. The distribution result is illustrated through numerical experiments, one of which is inspired by the finite element approximation of a system of differential equations.

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Abstract The least squares vorticity stabilization (LSVS), proposed in N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzmán, A. Linke, and C. Merdon (“A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation,” SIAM J. Numer. Anal., vol. 59, no. 5, pp. 2746–2774, 2021) for the Scott–Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor–Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott–Vogelius and Taylor–Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott–Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method.

Abstract We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest G(u ⋆) with a linear goal functional G and the solution u ⋆ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

Abstract We study the stability and convergence properties of a semi-implicit time stepping scheme for the incompressible Navier–Stokes equations with variable density and viscosity. The density is assumed to be approximated in a way that conserves the minimum-maximum principle. The scheme uses a fractional time-stepping method and the momentum, which is equal to the product of the density and velocity, as a primary unknown. The semi-implicit algorithm for the coupled momentum-pressure is shown to be conditionally stable and the velocity is shown to converge in L 2 norm with order one in time. Numerical illustrations confirm that the algorithm is stable and convergent under classic CFL condition even for sharp density profiles.

Abstract In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d , d = 2, 3. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ⩾ 1 for the velocity and polynomials of degree k − 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L 2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

Abstract We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant (or equi-variant) under permutations and rotations. This task arises in the evaluation of linear models as well as equivariant neural network models of many-particle systems. The theoretical bottleneck is the contraction of a high-dimensional symmetric and sparse tensor with a specific sparsity pattern that is directly related to the symmetries imposed on the polynomial. The sparsity of this tensor makes it challenging to construct a highly efficient evaluation scheme. Bachmayr et al. (“Polynomial approximation of symmetric functions,” Math. Comp., vol. 93, pp. 811–839, 2024) and Lysogorskiy et al. (“Performant implementation of the atomic cluster expansion (pace): application to copper and silicon,” npj Comput. Mater., vol. 7, Art. no. 97, 2021) introduced a recursive evaluation strategy that relied on a number of heuristics, but performed well in tests. In the present work, we propose an explicit construction of such a recursive evaluation strategy and show that it is in fact optimal in the limit of infinite polynomial degree.

Article Corrigendum to: Numerical analysis for a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport was published on June 1, 2024 in the journal Journal of Numerical Mathematics (volume 32, issue 2).