Polynomial Space Research Articles

A nonconforming immersed virtual element method for elliptic interface problems

This paper presents the lowest-order nonconforming immersed virtual element method for solving elliptic interface problems on unfitted polygonal meshes. The local discrete space on each interface mesh element consists of the solutions of local interface problems with Neumann boundary conditions, and the elliptic projection is modified so that its range is the space of broken linear polynomials satisfying the interface conditions. We derive optimal error estimates in the broken H1-norm and L2-norm, under the piecewise H2-regulartiy assumption. In our scheme, the mesh assumptions for error analysis allow small cut elements. Several numerical experiments are provided to confirm the theoretical results.

Improving the third-order WENO schemes by using exponential polynomial space with a locally optimized shape parameter

Improving the third-order WENO schemes by using exponential polynomial space with a locally optimized shape parameter

Polyhedral control-net splines for analysis

Generalizing tensor-product splines to smooth functions whose control nets can outline more general topological polyhedra, bi-cubic polyhedral-net splines form a first-order differentiable piecewise polynomial space. Each polyhedral control net node is associated with one bi-cubic function. A polyhedral control net admits grid-, star-, n-gon-, polar- and three types of T-junction configurations. Analogous to tensor-product splines, polyhedral-net splines can both model curved geometry and represent higher-order functions on the geometry. This paper explores the use of polyhedral-net splines for solving elliptic partial differential equations on curved smooth free-form surfaces without additional meshing.

Three types of reproducing kernel Hilbert spaces of polynomials

In this paper we will investigate reproducing kernel Hilbert spaces of polynomials of degree at most n with three different inner products: given by an integral with a weight, given by the sum of products of values of a polynomial at n + 1 points and given by the sum of products of coefficients of the same power. In the first case we will show that the reproducing kernel depends continuously on deformation of an inner product in a precisely defined sense. In the second and third case we will give a formula for the reproducing kernel.

The Newton polytope of the Morse discriminant of a univariate polynomial

In this paper we compute the Newton polytope MA of the Morse discriminant in the space of univariate polynomials with a given support set A. Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of MA.

Convex computation of maximal Lyapunov exponents

We describe an approach for finding upper bounds on an ODE dynamical system’s maximal Lyapunov exponent (LE) among all trajectories in a specified set. A minimisation problem is formulated whose infimum is equal to the maximal LE, provided that trajectories of interest remain in a compact set. The minimisation is over auxiliary functions that are defined on the state space and subject to a pointwise inequality. In the polynomial case—i.e. when the ODE’s right-hand side is polynomial, the set of interest can be specified by polynomial inequalities or equalities, and auxiliary functions are sought among polynomials—the minimisation can be relaxed into a computationally tractable polynomial optimisation problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimisation problems of increasing computational cost whose infima converge from above to the maximal LE, at least when the set of interest is compact. For illustration, we carry out such polynomial optimisation computations for two chaotic examples: the Lorenz system and the Hénon–Heiles system. The computed upper bounds converge as polynomial degrees are raised, and in each example we obtain a bound that is sharp to at least five digits. This sharpness is confirmed by finding trajectories whose leading Lyapunov exponents approximately equal the upper bounds.

On Primary Decomposition of Hermite Projectors

An ideal projector on the space of polynomials C[x]=C[x1,…,xd] is a projector whose kernel is an ideal in C[x]. Every ideal projector P can be written as a sum of ideal projectors P(k) such that the intersection of their kernels kerP(k) is a primary decomposition of the ideal kerP. In this paper, we show that P is a limit of Lagrange projectors if and only if each P(k) is. As an application, we construct an ideal projector P whose kernel is a symmetric ideal, yet P is not a limit of Lagrange projectors.

An operator splitting Legendre-tau spectral method for Maxwell’s equations with nonlinear conductivity in two dimensions

In the paper, an operator splitting Legendre-tau spectral method is proposed to solve Maxwell’s equations with nonlinear conductivity in two dimensions. By the implicit–explicit numerical scheme, the linear and nonlinear parts are treated in different ways. The linear terms are treated by the Legendre-tau spectral method implicitly, and the nonlinear terms are treated by some collocation methods explicitly. The scheme uses polynomial spaces of different degrees to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. The splitting leap frog Crank–Nicolson method is applied in time discretization, which is a three-level scheme with the nonlinear term being treated explicitly in intermediate level. With the application of inverse inequality, for the Legendre interpolation operator, optimal L2-error estimates are achieved for the proposed scheme. In numerical examples, the nonlinear terms are computed at Chebyshev–Gauss–Lobatto points by the fast Legendre transform. Numerical results validate the efficiency and spectral accuracy of the scheme.

On polynomial Trefftz spaces for the linear time-dependent Schrödinger equation

We study the approximation properties of complex-valued polynomial Trefftz spaces for the (d+1)-dimensional linear time-dependent Schrödinger equation. More precisely, we prove that for the space–time Trefftz discontinuous Galerkin variational formulation proposed by Gómez and Moiola (2022), the same h-convergence rates as for polynomials of degree p in (d+1) variables can be obtained in a mesh-dependent norm by using a space of Trefftz polynomials of anisotropic degree. For such a space, the dimension is equal to that of the space of polynomials of degree 2p in d variables, and bases are easily constructed.

Basic Properties of Solutions of Boundary Value Problem for the Dirac-Wave Operator

We study local and global properties of solutions of the boundary value problem for the Dirac wave operator. This includes explicit formulas, domain of dependence, range of influence, finite propagation speed and energy estimates. We also introduce the space of Dirac wave homogeneous polynomials and the corresponding Dirac-wave conjugate functions.

On the homology of spaces of equivariant maps

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main motivating example, we calculate the rational homology groups of spaces of even and odd maps [Formula: see text], [Formula: see text], or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps [Formula: see text] of increasing degree. Also, we calculate the homology groups of spaces of [Formula: see text]-equivariant maps of odd-dimensional spheres for any [Formula: see text]. In auxiliary calculations, we find the homology groups of configuration spaces of projective and lens spaces with coefficients in several local systems.

Gaussian type quadrature rules related to the oscillatory modification of the generalized Laguerre weight functions

In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions wα(x)=xαe−x, α>−1, on (0,+∞) by the oscillatory factor exp(iζx), where ζ>0. For calculation of such integrals for analytic real-valued functions in D={z∈ℂ∣Rez≥0,Imz≥0} we construct polynomials orthogonal with respect to the linear functional L:P→ℂ, given by L[p]=∫0+∞p(x)wα(x)exp(iζx)dx, where P is a linear space of all algebraic polynomials, as well as the corresponding quadrature rules of Gaussian type. The existence of such orthogonal polynomials and the corresponding three-term recurrence relations are proved. A few numerical examples are presented, including computation of the Cauchy principal value integrals.

Irreducibility of periodic curves in cubic polynomial moduli space

AbstractIn the moduli space of complex cubic polynomials with a marked critical point, given any , we prove that the loci formed by polynomials with the marked critical point periodic of period is an irreducible curve. Thus, answering a question posed by Milnor in the 1990s.

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Total positivity and accurate computations with Gram matrices of Said‐Ball bases

AbstractIn this article, it is proved that Gram matrices of totally positive bases of the space of polynomials of a given degree on a compact interval are totally positive. Conditions to guarantee computations to high relative accuracy with those matrices are also obtained. Furthermore, a fast and accurate algorithm to compute the bidiagonal factorization of Gram matrices of the Said‐Ball bases is obtained and used to compute to high relative accuracy their singular values and inverses, as well as the solution of some linear systems associated with these matrices. Numerical examples are included.

Symmetric cubic laminations

To investigate the degree d d connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied σ d \sigma _d -invariant laminations, where σ d \sigma _d is the d d -tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f ( z ) = z 2 + c f(z) = z^2 +c . In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials f λ ( z ) = z 3 + λ 2 z f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s C L C_sCL together with the induced factor space S / C s C L \mathbb {S}/C_sCL of the unit circle S \mathbb {S} . As will be verified in the third paper of the series, S / C s C L \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

A pressure-robust mixed finite element method for the coupled Stokes–Darcy problem

A pressure-robust numerical method for the coupled Stokes–Darcy problem is proposed. We use the continuous piecewise linear polynomial space combined with the lowest order Raviart–Thomas space to discretize the Stokes equation in the fluid region and the lowest order Raviart–Thomas elements to discretize the Darcy equation in the porous media domain. We also perform a divergence-free reconstruction of the test function for the right-hand term. Our method has good properties of convergence, pressure-robustness and local mass conservation in the sense of projection. A discrete inf-sup condition and error estimates for the numerical scheme are demonstrated, the error of velocity is independent of viscosity and pressure. Finally, the theoretical analysis is validated with several examples, demonstrating the superiority of the pressure-robust scheme.

The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the smallest number of lines in $\mathbb{R}^d$ whose union contains a crossing-free straight-line drawing of $G$. For $d=2$, the graph $G$ must be planar. Similarly, let $\rho^2_3(G)$ denote the smallest number of planes in $\mathbb{R}^3$ whose union contains a crossing-free straight-line drawing of $G$. We investigate the complexity of computing these three parameters and obtain the following hardness and algorithmic results. For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is $\exists\mathbb{R}$-complete. Since $NP \subseteq \exists\mathbb{R}$, deciding $\rho^1_d(G)\le k$ is $NP$-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. Since $\exists\mathbb{R} \subseteq PSPACE$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. We prove that deciding whether $\rho^2_3(G)\le k$ is $NP$-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $P=NP$.

Approximating Lipschitz and continuous functions by polynomials; Jackson’s theorem

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be ), denote by C (I) the space of continuous functions on I, and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that . Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write

Explicit consistency conditions for fully symmetric cubature on the tetrahedron

A novel fully symmetric basis is derived for the S4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_4$$\\end{document}-invariant polynomial space, by using symmetric polynomials and invariant theory. This new basis enables deriving explicitly the consistency conditions for non-overdeterminedness of moment equations in the case of fully symmetric cubature rules on the tetrahedron. Solving the corresponding linear integer programming problem, optimal and quasi-optimal rule structures are derived. Explicit formulas to calculate the estimated lower bounds in the number of integration points are also given. Additionally, the new basis is of practical interest in calculating specific cubature rules, since it allows decomposing the moment equations into a series of successively independent smaller subsystems, which can be exploited in designing more efficient solution methods. Solving the moment equations analytically we obtain several interesting new results.