Polynomials Of Arbitrary Degree Research Articles

Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus

Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. Harmonic forms are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular cohomology class is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed piecewise polynomial differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed polynomial forms, that is, arbitrary degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.

On the matrices in B‐spline collocation methods for Riesz fractional equations and their spectral properties

AbstractIn this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B‐spline collocation method. For an arbitrary polynomial degree , we show that the resulting coefficient matrices possess a Toeplitz‐like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, the given matrices are ill‐conditioned both in the low and high frequencies for large . More precisely, in the fractional scenario the symbol vanishes at 0 with order , the fractional derivative order that ranges from 1 to 2, and it decays exponentially to zero at for increasing at a rate that becomes faster as approaches 1. This translates into a mitigated conditioning in the low frequencies and into a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B‐spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B‐splines. Finally, we perform a numerical study of the approximation behavior of polynomial B‐spline collocation. This study suggests that, in line with nonfractional diffusion problems, the approximation order for smooth solutions in the fractional case is for even , and for odd .

Weak Galerkin coupled with conforming finite element method for hybrid-dimensional fracture model

In this paper, we propose the weak Galerkin coupled with conforming finite element method to solve the hybrid-dimensional fracture problem, which couples n-dimensional Darcy flow in the porous matrix with (n−1)-dimensional flow in the fracture. The generalized Robin type interface conditions present in the coupling scheme. We derive the optimal error estimates for L2 and energy norms with arbitrary degree polynomials. The error rate obtained by numerical examples agrees with the analysis results on both polygonal mesh and nonmatching mesh. Results of numerical examples also show that the proposed scheme is robust to the parameters in the reduced fracture model.

Математическое моделирование динамики тепловых ударных волн в нелинейных локально-неравновесных средах

We performed a mathematical simulation of heat transfer in a local non-equilibrium medium whose transfer characteristics are functions of the temperature distribution. A homogeneous polynomial of arbitrary degree represents nonlinearities in thermal conductivity and thermal diffusivity. The mathematical model consists of a hyperbolic nonlinear heat transfer wave equation, initial conditions and nonlinear boundary conditions of the second and first kind. To solve this problem, we used a conservative homogeneous finite-difference scheme along the upper time grid line (implicitly). We then used the tridiagonal matrix algorithm of the second order in the spatial variable and of the first order in time to solve the resulting system of linearised algebraic equations. A periodic series of rectangular temperature or heat flux pulses form the boundary conditions of the first and second kind. Computation results reveal ultimate propagation rates of temperature and heat fronts featuring pronounced first-kind discontinuities with attenuating magnitudes. As the process unfolds, the initial pulses heat the region between the boundary and the heat wave front, while the subsequent pulses traverse this region at a higher velocity due to thermal diffusivity being a function of temperature, their fronts "catching up" with the previous fronts, increasing the discontinuity magnitude at the initial pulse front, that is, forming a thermal shock wave front similar to that of a shock wave in gas dynamics. We obtained such thermal shock waves for boundary conditions of both the first and the second kind. We also analysed kinematic and dynamic characteristics of thermal waves

Robust A Posteriori Error Analysis for Rotation-Based Formulations of the Elasticity/Poroelasticity Coupling

We develop the a posteriori error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretizations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lamé constants); they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behavior predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the a posteriori error estimators.

Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

We formulate and analyze a goal-oriented adaptive finite element method for a semilinear elliptic PDE and a linear goal functional. The discretization is based on finite elements of arbitrary (but fixed) polynomial degree and involves a linearized dual problem. The linearization is part of the proposed algorithm, which employs a marking strategy different to that of standard adaptive finite element methods. Moreover, unlike the well-known dual-weighted residual strategy, the analyzed error estimators are completely computable. We prove linear convergence and, for the first time in the context of goal-oriented adaptivity for nonlinear PDEs, optimal algebraic convergence rates. In particular, the analysis does not require a sufficiently fine initial mesh.

How to prove optimal convergence rates for adaptive least-squares finite element methods∗

Abstract The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53 (1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.

Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

In 2018, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron–Frobenius operators L_s , an idea known in varying degrees of generality for many years. Although L_s is not compact in the setting we consider, it possesses a strictly positive C^m eigenfunction v_s with eigenvalue R(L_s) for arbitrary m and all other points z in the spectrum of L_s satisfy |z| \le b for some constant b < R(L_s) . Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which R(L_s) =1 . This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree r . Using an extension of the Perron theory of positive matrices to matrices that map a cone K to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s , we give rigorous upper and lower bounds for the Hausdorff dimension s_* , and these bounds converge rapidly to s_* as the mesh size decreases and/or the polynomial degree increases.

The image of multilinear polynomials evaluated on 3 × 3 upper triangular matrices

Abstract We describe the images of multilinear polynomials of arbitrary degree evaluated on the 3×3 upper triangular matrix algebra over an infinite field.

Convergence of Adaptive Discontinuous Galerkin and $$C^0$$-Interior Penalty Finite Element Methods for Hamilton–Jacobi–Bellman and Isaacs Equations

We prove the convergence of adaptive discontinuous Galerkin and \(C^0\)-interior penalty methods for fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including Poincaré and trace inequalities, a proof of the density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results.

(In)security of concrete instantiation of Lin17’s functional encryption scheme from noisy multilinear maps

Functional encryption (FE) is a novel cryptographic paradigm. In comparison to conventional encryption schemes, FE allows producing secret keys $$sk_f$$ corresponding to a function f that decrypt encryptions of $$x_0$$ to $$f(x_0)$$ . Recently, Lin proposed FE for arbitrary degree polynomials from the SXDH assumption to an exact multilinear map (CRYPTO’17). However, there is no concrete instantiation of the scheme in the absence of an exact multilinear map. Although Lin’s FE can be instantiated by noisy multilinear maps such as the GGH13, CLT13, and GGH15 schemes, the security of FE instantiated by noisy multilinear maps is unclear. In this paper, we point out the weakness of the Lin’s FE when it is instantiated by well-known candidates of noisy multilinear maps. In other words, we present a polynomial time attack of the FE on each noisy multilinear map. In the proposed method, our attack captures Lin’s FE for arbitrary degree polynomials instantiated by GGH13 and CLT13 and is also applicable to FE for polynomials of degree $$O(\log _2 \lambda )$$ when instantiated by GGH15 under the current parameters where $$\lambda $$ is the security parameter.

Ck and C0hp-finite elements on d-dimensional meshes with arbitrary hanging nodes

In this paper, the construction of Ck basis functions is proposed for paraxial d-dimensional rectangular meshes with arbitrary hanging nodes and arbitrary polynomial degree distributions. The construction is based on the large-support approach introduced in [1] for C0 basis functions in 2D and uses hierarchical tensor-product shape functions which combine Hermite shape functions with Gegenbauer polynomials enabling the support of the basis functions to be independent of k (in contrast to basis functions based on B-spline approaches). Moreover, these shape functions allow for an efficient recursive computation of the constraints coefficients in the application of constrained approximation for hanging nodes without the need for collocation. An appropriate indexing of the shape functions is introduced in order to prove the differentiability properties of the basis functions. The construction is also suitable for the extension to C0 finite elements on meshes which are not necessarily rectangular. In particular, the orientation problem resulting from differently oriented edges or faces can be appropriately treated within this extension. Numerical examples illustrate the feasibility of the proposed approach. Moreover, some aspects concerning the condition number of the system matrix resulting from the discretization of Poisson's problem are discussed.

An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis functions yields cheaply invertible block-diagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence.

Duality in Finite Element Exterior Calculus and Hodge Duality on the Sphere

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite element exterior calculus began with the $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k and $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k and $$\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}$$ P ˚ r + k + 1 - Λ n - k and between $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k and $$\mathring{{\mathcal {P}}}_{r+k}\varLambda ^{n-k}$$ P ˚ r + k Λ n - k . In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard n-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k , $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k , $$\mathring{{\mathcal {P}}}_r\varLambda ^k$$ P ˚ r Λ k and $$\mathring{{\mathcal {P}}}_r^-\varLambda ^k$$ P ˚ r - Λ k spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $${\mathcal {P}}_r \varLambda ^k$$ P r Λ k and $$\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}$$ P ˚ r + k + 1 - Λ n - k and between $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k and $$\mathring{{\mathcal {P}}}_{r+k} \varLambda ^{n-k}$$ P ˚ r + k Λ n - k . These isomorphisms can be implemented via a simple computation, which we illustrate with examples.

Collective marking for arbitrary order adaptive least-squares finite element methods with optimal rates

The collective marking strategy with alternative refinement-indicators in adaptive mesh-refining of least-squares finite element methods (LSFEMs) has recently been shown to lead to optimal convergence rates in Carstensen (2020). The proofs utilize explicit identities for the lowest-order Raviart–Thomas and the Crouzeix–Raviart finite elements. This paper generalizes those results to arbitrary polynomial degree and mixed boundary conditions with some novel arguments. The analysis is outlined for the Poisson equation in 3D with mixed boundary conditions.

Central discontinuous Galerkin methods on overlapping meshes for wave equations

In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise Pk elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k ≥ 0. In particular, we adopt the techniques in Liu et al. (SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.

Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations.

Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations x ¨ + f ( x ) x ˙ + x = 0 , being f ( x ) a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified.

A-Posteriori-Steered $p$-Robust Multigrid with Optimal Step-Sizes and Adaptive Number of Smoothing Steps

We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree $p ≥ 1$. Our solver employs zero pre- and one post-smoothing by the overlapping Schwarz (block-Jacobi) method and features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and level-wise and patch-wise error reductions. We show the two following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree $p$; and the estimator represents a two-sided $p$-robust bound on the algebraic error. The $p$-robustness results are obtained by carefully applying the results of Schoberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1–24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of Xu et al. [Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659]. We consider quasi-uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal $H^1$-regularity and complete independence for $H^2$-regularity. We also present a simple and effective way for the solver to adaptively choose the number of post-smoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm $p$-robustness and show the benefits of the adaptive number of smoothing steps.

A Stabilizer-Free, Pressure-Robust, and Superconvergence Weak Galerkin Finite Element Method for the Stokes Equations on Polytopal Mesh

In this paper, we propose a new stabilizer-free and pressure-robust weak Galerkin finite element method for the Stokes equations with superconvergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal order for velocity in both the energy norm and the $L^2$-norm and for pressure in the $L^2$-norm are proved in our proposed scheme. The $H$(div)-preserving operator has been constructed based on the polygonal mesh for arbitrary polynomial degrees and employed in the body source assembling to break the locking phenomenon induced by poor mass conservation in the classical discretization. Moreover, the velocity error in our proposed scheme is proved to be independent of pressure and thus we confirm the pressure-robustness. For Stokes simulation, our proposed scheme only modifies the body source assembling but keeps the same stiffness matrix. Four numerical experiments are conducted to validate the convergence results and robustness.

A sequence of polynomials with optimal condition number

We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.