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We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally $C^4$-smooth isogeometric spline space [25] to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the $L^2$-norm as well as to equivalents of the $H^s$-seminorms for $1 \leq s \leq 4$.

Implicit models for magnetic coenergy have been proposed by Pera et al. to describe the anisotropic nonlinear material behavior of electrical steel sheets. This approach aims at predicting magnetic response for any direction of excitation by interpolating measured of B--H curves in the rolling and transverse directions. In an analogous manner, an implicit model for magnetic energy is proposed. We highlight some mathematical properties of these implicit models and discuss their numerical realization, outline the computation of magnetic material laws via implicit differentiation, and discuss the potential use for finite element analysis in the context of nonlinear magnetostatics.

Quadratic NURBS-based discretizations of the Galerkin method suffer from volumetric locking when applied to nearly-incompressible linear elasticity. Volumetric locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of normal stresses. Continuous-assumed-strain (CAS) elements have been recently introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we propose two generalizations of CAS elements (named CAS1 and CAS2 elements) to overcome volumetric locking in quadratic NURBS-based discretizations of nearly-incompressible linear elasticity. CAS1 elements linearly interpolate the strains at the knots in each direction for the term in the variational form involving the first Lam\'e parameter while CAS2 elements linearly interpolate the dilatational strains at the knots in each direction. For both element types, a displacement vector with C1 continuity across element boundaries results in assumed strains with C0 continuity across element boundaries. In addition, the implementation of the two locking treatments proposed in this work does not require any additional global or element matrix operations such as matrix inversions or matrix multiplications. The locking treatments are applied at the element level and the nonzero pattern of the global stiffness matrix is preserved. The numerical examples solved in this work show that CAS1 and CAS2 elements, using either two or three Gauss-Legrendre quadrature points per direction, are effective locking treatments since they not only result in more accurate displacements for coarse meshes, but also remove the spurious oscillations of normal stresses.

This work is concerned with the analysis of a space-time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic-elastic media. The mathematical model consists of the low-frequency Biot's equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling, suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is then coupled with a dG time integration scheme, resulting in a full space-time dG discretization. We present the stability analysis for both the continuous and the semidiscrete formulations, and we derive error estimates for the semidiscrete formulation in a suitable energy norm. The method is applied to a wide set of numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.

We propose a new method for the construction of layer-adapted meshes for singularly perturbed differential equations (SPDEs), based on mesh partial differential equations (MPDEs) that incorporate \emph{a posteriori} solution information. There are numerous studies on the development of parameter robust numerical methods for SPDEs that depend on the layer-adapted mesh of Bakhvalov. In~\citep{HiMa2021}, a novel MPDE-based approach for constructing a generalisation of these meshes was proposed. Like with most layer-adapted mesh methods, the algorithms in that article depended on detailed derivations of \emph{a priori} bounds on the SPDE's solution and its derivatives. In this work we extend that approach so that it instead uses \emph{a posteriori} computed estimates of the solution. We present detailed algorithms for the efficient implementation of the method, and numerical results for the robust solution of two-parameter reaction-convection-diffusion problems, in one and two dimensions. We also provide full FEniCS code for a one-dimensional example.

We study the numerical approximation of a coupled hyperbolic-parabolic system by a family of discontinuous Galerkin space-time finite element methods. The model is rewritten as a first-order evolutionary problem that is treated by the unified abstract solution theory of R.\ Picard. To preserve the mathematical structure of the evolutionary equation on the fully discrete level, suitable generalizations of the distribution gradient and divergence operators on broken polynomial spaces on which the discontinuous Galerkin approach is built on are defined. Well-posedness of the fully discrete problem and error estimates for the discontinuous Galerkin approximation in space and time are proved.

Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws such as the well-known policy iteration. In this manuscript, the focus is on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman or composition operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup, in a specially weighted $\mathrm{L}^p$-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay which justifies finite rank approximations with numerical methods. The potential benefit for numerical computations will be demonstrated with two short examples.

Regularized generalized canonical correlation analysis (RGCCA) is a generalization of regularized canonical correlation analysis to three or more sets of variables, which is a component-based approach aiming to study the relationships between several sets of variables. Sparse generalized canonical correlation analysis (SGCCA) (proposed in Tenenhaus et al. (2014)), combines RGCCA with an `1-penalty, in which blocks are not necessarily fully connected, makes SGCCA a flexible method for analyzing a wide variety of practical problems, such as biology, chemistry, sensory analysis, marketing, food research, etc. In Tenenhaus et al. (2014), an iterative algorithm for SGCCA was designed based on the solution to the subproblem (LM-P1 for short) of maximizing a linear function on the intersection of an `1-norm ball and a unit `2-norm sphere proposed in Witten et al. (2009). However, the solution to the subproblem (LM-P1) proposed in Witten et al. (2009) is not correct, which may become the reason that the iterative algorithm for SGCCA is slow and not always convergent. For this, we first characterize the solution to the subproblem LM-P1, and the subproblems LM-P2 and LM-P3, which maximize a linear function on the intersection of an `1-norm sphere and a unit `2-norm sphere, and an `1-norm ball and a unit `2-norm sphere, respectively. Then we provide more efficient block coordinate descent (BCD) algorithms for SGCCA and its two variants, called SGCCA-BCD1, SGCCA-BCD2 and SGCCA-BCD3, corresponding to the subproblems LM-P1, LM-P2 and LM-P3, respectively, prove that they all globally converge to their stationary points. We further propose gradient projected (GP) methods for SGCCA and its two variants when using the Horst scheme, called SGCCA-GP1, SGCCA-GP2 and SGCCA-GP3, corresponding to the subproblems LM-P1, LM-P2 and LM-P3, respectively, and prove that they all

We propose to approximate a (possibly discontinuous) multivariate function f (x) on a compact set by the partial minimizer arg miny p(x, y) of an appropriate polynomial p whose construction can be cast in a univariate sum of squares (SOS) framework, resulting in a highly structured convex semidefinite program. In a number of non-trivial cases (e.g. when f is a piecewise polynomial) we prove that the approximation is exact with a low-degree polynomial p. Our approach has three distinguishing features: (i) It is mesh-free and does not require the knowledge of the discontinuity locations. (ii) It is model-free in the sense that we only assume that the function to be approximated is available through samples (point evaluations). (iii) The size of the semidefinite program is independent of the ambient dimension and depends linearly on the number of samples. We also analyze the sample complexity of the approach, proving a generalization error bound in a probabilistic setting. This allows for a comparison with machine learning approaches.

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the geometry of the domain.