Abstract We propose a positivity preserving finite element discretization for the nonlinear Gross–Pitaevskii eigenvalue problem. The method employs mass lumping techniques, which allow to transfer the uniqueness up to sign and positivity properties of the continuous ground state to the discrete setting. We further prove that every non-negative discrete excited state up to sign coincides with the discrete ground state. This allows one to identify the limit of fully discretized gradient flows, which are typically used to compute the discrete ground state, and thereby establish their global convergence. Furthermore, we perform a rigorous a priori error analysis of the proposed non-standard finite element discretization, showing optimal orders of convergence for all unknowns. Numerical experiments illustrate the theoretical results of this paper.

Abstract In this work, we investigate the boundary controllability of time-discrete parabolic systems, uniformly with respect to the discretization parameter. To establish our main results, we adapt and extend the moment method first introduced by Fattorini and Russell, to the time-discrete setting. While this method has proven effective in the continuous framework and in the space-discrete case, its adaptation to time-discrete systems is challenging due to the fact that the more accurate results in the field are based on several complex analysis tools. To overcome this, we introduce a new alternative proof for constructing biorthogonals to (generalized) exponential functions in the continuous setting, based on the original proof by Fattorini and Russell which avoids the use of these tools, and taking ideas from the block moment method introduced by Benabdallah, Morancey and the first author of this work. We then manage to adapt this strategy to the discrete setting, enabling the construction and estimation of biorthogonal families for some time-discrete functions that play the same role as the exponentials at the discrete level. Our results show that these biorthogonals can be uniformly estimated for a finite portion of the spectrum, determined by the discretization parameter $$\tau$$ , and that converges to the whole spectrum as $$\tau$$ goes to zero. Using this tool, we prove a relaxed null-controllability result for discrete parabolic systems. It says that there exists a bounded sequence of time discrete controls that makes the solution reach a target at final time tending to zero exponentially fast as the discretization parameter $$\tau$$ goes to zero. We will also study the effects of the existence of a minimal null-control time and how this phenomenon translates into the discrete world.

Abstract In this note, we introduce a novel stabilization mechanism for Cut Finite Element Methods (CutFEM), generalizing previous ghost penalty techniques in two key aspects: (1) the choice of stabilized quantities and (2) the selection of elements involved in stabilization. This approach notably allows for flexible and precise definitions of the stabilized quantities, including various functionals associated with the discrete solution, such as finite element degrees of freedom. We demonstrate that the kernel of our proposed ghost penalty operator defines a finite element space characterized by discrete extensions, closely related to those previously presented in Burman et al. (Numer Math 152(2):331–369, 2022).

Abstract This work describes and analyzes the domain derivative for a time-dependent acoustic scattering problem. We study the nonlinear operator that maps a sound-soft scattering object to the solution of the time-dependent wave equation evaluated at a finite number of points away from the obstacle. The Fréchet derivative of this operator with respect to variations of the scatterer coincides with point evaluations of the temporal domain derivative. The latter is the solution to another time-dependent scattering problem, for which a well-posedness result is shown under sufficient temporal regularity of the incoming wave. Applying convolution quadrature to this scattering problem gives a stable and provably convergent semi-discretization in time, provided that the incoming wave is sufficient regular. Using the discrete domain derivative in a Gauss–Newton method, we describe an efficient algorithm to reconstruct the boundary of an unknown scattering object from time domain measurements in a few points away from the boundary. Numerical examples for the acoustic wave equation in two dimensions demonstrate the performance of the method.

Abstract We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval $$(-1,1)$$ ( - 1 , 1 ) . We assume that the given source term and reaction coefficient are analytic in $$[-1,1]$$ [ - 1 , 1 ] . The expression rate bounds in Sobolev norms in terms of the NN size are robust, i.e. uniform with respect to the singular perturbation parameter $$\varepsilon \in (0,1]$$ ε ∈ ( 0 , 1 ] for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and $$\tanh $$ tanh - and sigmoid-activated NNs. The latter activations can represent “exponential boundary layer solution features” explicitly, in the last hidden layer of the DNN, i.e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. All DNN architectures allow robust exponential solution expression in so-called ‘energy’ as well as in ‘balanced’ Sobolev norms, for analytic input data.

Abstract We extend several celebrated methods in classical analysis for summing series of complex numbers to series of complex matrices. These include the summation methods of Abel, Borel, Cesàro, Euler, Lambert, Nörlund, and Mittag-Leffler, which are frequently used to sum scalar series that are divergent in the conventional sense. One feature of our matrix extensions is that they are fully noncommutative generalizations of their scalar counterparts—not only is the scalar series replaced by a matrix series, positive weights are replaced by positive definite matrix weights, order on $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> replaced by Loewner order, exponential function replaced by matrix exponential function, etc. We will establish the regularity of our matrix summation methods, i.e., when applied to a matrix series convergent in the conventional sense, we obtain the same value for the sum. Our second goal is to provide numerical algorithms that work in conjunction with these summation methods. We discuss how the block and mixed-block summation algorithms, the Kahan compensated summation algorithm, may be applied to matrix sums with similar roundoff error bounds. These summation methods and algorithms apply not only to power or Taylor series of matrices but to any general matrix series including matrix Fourier and Dirichlet series. We will demonstrate the utility of these summation methods: establishing a Fejér’s theorem and alleviating the Gibbs phenomenon for matrix Fourier series; extending the domains of matrix functions and accurately evaluating them; enhancing the matrix Padé approximation and Schur–Parlett algorithms; and more.

Abstract In this paper error analysis for finite element discretizations of Dirichlet boundary control problems is developed. For the first time, optimal discretization error estimates are established in the case of three dimensional polyhedral and convex domains. The convergence rates solely depend on the size of largest interior edge angle. These results are comparable to those for the two dimensional case. However, the approaches from the two dimensional setting are not directly extendable such that new techniques have to be used. The theoretical results are confirmed by numerical experiments.

Abstract We consider a systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer-solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows us to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations, which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.