Piecewise Polynomials Of Degree Research Articles

Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system

This paper proposes numerical developments based on polynomial chaos (PC) expansions to process stochastic eigenvalue problems efficiently. These developments are applied to the problem of linear stability calculations for a simplified brake system: the stability of a finite element model of a brake is investigated when its friction coefficient or the contact stiffness are modeled as random parameters. Getting rid of the statistical point of view of the PC method but keeping the principle of a polynomial decomposition of eigenvalues and eigenvectors, the stochastic space is decomposed into several elements to realize a low degree piecewise polynomial approximation of these quantities. An approach relying on continuation principles is compared to the classical dichotomy method to build the partition. Moreover, a criterion for testing accuracy of the decomposition over each cell of the partition without requiring evaluation of exact eigenmodes is proposed and implemented.Several random distributions are tested, including a uniform-like law for description of friction coefficient variation. Results are compared to Monte Carlo simulations so as to determine the method accuracy and efficiency. Some general rules relative to the influence of the friction coefficient or the contact stiffness are also inferred from these calculations.

Optimal Control of Switched Systems based on Bezier Control Points

This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a pre-specified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. A Bezier control points method is applied for solving an optimal control problem which is supervised by a switched dynamic system. Two steps of approximation exist here. First, the time interval is divided into sub-intervals. Second, the trajectory and control functions are approximatedby Bezier curves in each subinterval. Bezier curves have been considered as piecewise polynomials of degree , then they will be determined by control points on any subinterval. The optimal control problem is there by converted into a nonlinear programming problem (NLP), which can be solved by known algorithms. However in this paper the MATLAB optimization routine FMINCON is used for solving resulting NLP. Index Terms—switched dynamical system, Bezier control points, optimal control.

Superconvergent Discontinuous Galerkin Methods for Linear Non-selfadjoint and Indefinite Elliptic Problems

Based on Cockburn et al. (Math. Comp. 78:1---24, 2009), superconvergent discontinuous Galerkin methods are identified for linear non-selfadjoint and indefinite elliptic problems. With the help of an auxiliary problem which is the discrete version of a linear non-selfadjoint elliptic problem in divergence form, optimal error estimates of order k+1 in L 2-norm for the potential and the flux are derived, when piecewise polynomials of degree k?1 are used to approximate both potential and flux variables. Using a suitable post-processing of the discrete potential, it is then shown that the resulting post-processed potential converges with order k+2 in L 2-norm. The article is concluded with a numerical experiment which confirms the theoretical results.

Collocation methods for solving multivariable integral equations of the second kind

In a recent paper (Allouch, in press) [5] on one dimensional integral equations of the second kind, we have introduced new collocation methods. These methods are based on an interpolatory projection at Gauss points onto a space of discontinuous piecewise polynomials of degree r which are inspired by Kulkarni’s methods (Kulkarni, 2003) [10], and have been shown to give a 4r+4 convergence for suitable smooth kernels. In this paper, these methods are extended to multi-dimensional second kind equations and are shown to have a convergence of order 2r+4. The size of the systems of equations that must be solved in implementing these methods remains the same as for Kulkarni’s methods. A two-grid iteration convergent method for solving the system of equations based on these new methods is also defined.

Error analysis of a modified discontinuous Galerkin recovery scheme for diffusion problems

A theoretical error analysis using standard Sobolev space energy arguments is furnished for a class of discontinuous Galerkin (DG) schemes that are modified versions of one of those introduced by van Leer and Nomura. These schemes, which use discontinuous piecewise polynomials of degree q, are applied to a family of one-dimensional elliptic boundary value problems. The modifications to the original method include definition of a recovery flux function via a symmetric L2-projection and the addition of a penalty or stabilization term. The method is found to have a convergence rate of O(hq) for the approximation of the first derivative and O(hq+1) for the solution. Computational results for the original and modified DG recovery schemes are provided contrasting them as far as complexity and cost. Numerical examples are given which exhibit sub-optimal convergence rates when the stabilization terms are omitted.

Superconvergence of Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws in One Space Dimension

In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.

Sampling and Reconstruction of Periodic Piecewise Polynomials Using Sinc Kernel

We address a problem of sampling and reconstructing periodic piecewise polynomials based on the theory for signals with a finite rate of innovation (FRI signals) from samples acquired by a sinc kernel. This problem was discussed in a previous paper. There was, however, an error in a condition about the sinc kernel. Further, even though the signal is represented by parameters, these explicit values are not obtained. Hence, in this paper, we provide a correct condition for the sinc kernel and show the procedure. The point is that, though a periodic piecewise polynomial of degree R is defined as a signal mapped to a periodic stream of differentiated Diracs by R+1 time differentiation, the mapping is not one-to-one. Therefore, to recover the stream is not sufficient to reconstruct the original signal. To solve this problem, we use the average of the target signal, which is available because of the sinc sampling. Simulation results show the correctness of our reconstruction procedure. We also show a sampling theorem for FRI signals with derivatives of a generic known function.

Error Estimates for an LDG Method Applied to Signorini Type Problems

In this paper we propose and analyze a Local Discontinuous Galerkin method for an elliptic variational inequality of the first kind that corresponds to a Poisson equation with Signorini type condition on part of the boundary. The method uses piecewise polynomials of degree one for the field variable and of degree zero or one for the approximation of its gradient. We show optimal convergence for the method and illustrate it with some numerical experiments.

Global convergence and local superconvergence of first-kind Volterra integral equation approximations

We present a comprehensive convergence analysis for discontinuous piecewise polynomial approximations of a first-kind Volterra integral equation with smooth convolution kernel, examining the attainable order of (super-) convergence in collocation (DC), quadrature discontinuous Galerkin (QDG) and full discontinuous Galerkin (DG) methods. We introduce new polynomial basis functions with properties that greatly simplify the convergence analysis for collocation methods. This also enables us to determine explicit formulae for the location of superconvergence points (i.e. discrete points at which the convergence order is one higher than the global bound) for \textbf{all} convergent collocation schemes. We show that a QDG method which is based on piecewise polynomials of degree $m$ and uses exactly $m+1$ quadrature points and nonzero quadrature weights is equivalent to a collocation scheme, and so its convergence properties are fully determined by the previous collocation analysis and they depend only on the quadrature point location (in particular, they are completely independent of the accuracy of the quadrature rule). We also give a complete analysis for QDG with more than $m+1$ quadrature points when the degree of precision (dop) is at least $2m+1$. The behaviour (but not the approximation) is the same as for a DG scheme when the dop is at least $2m+2$. Numerical test results confirm that the theoretical convergence rates are optimal.

A local projection type stabilization with exponential enrichments applied to one-dimensional advection–diffusion equations

We consider the local projection stabilization (LPS) for solving a singularly perturbed advection–diffusion two-point boundary value problem. In its classical one-level variant, the LPS uses polynomial bubble functions to enrich the standard finite element spaces of continuous, piecewise polynomial functions. As recently shown, the two-level approach can be considered also as a one-level method, however, with piecewise polynomial enrichments. Here, we study the question under which condition a linearly independent H 1 function can serve as an enrichment for the standard space of continuous, piecewise polynomials of degree r leading to the same type of error estimates for the solution as the original one- and two-level approaches. Moreover, in the constant coefficient case, we derive formulas for the user-chosen stabilization parameter which guarantee that the piecewise linear part of the solution becomes nodal exact. Finally, we choose exponential enrichments based on the asymptotic expansion of the solution and show by numerical tests that compared to the classical one-level variant of the LPS – a considerable improvement of the accuracy of the solution on non-layer adapted meshes can be achieved.

The two-level local projection stabilization as an enriched one-level approach

The two-level local projection stabilization is considered as a one-level approach in which the enrichments on each element are piecewise polynomial functions. The dimension of the enrichment space can be significantly reduced without losing the convergence order. On triangular meshes, for example, using continuous piecewise polynomials of degree r???1, only 2r???1 functions per macro-cell are needed for the enrichment compared to r 2 in the two-level approach. In case of the Stokes problem r???1 functions per macro-cell are already sufficient to guarantee stability and to preserve convergence order. On quadrilateral meshes the corresponding reduction rates are even higher. We give examples of "reduced" two-level approaches and study how the constant in the local inf-sup condition for the one-level and different two-level approaches, respectively, depends on the polynomial degree r.

Uniform-in-time superconvergence of HDG methods for the heat equation

We prove that the superconvergence properties of the hybridizable discontinuous Galerkin method for second-order elliptic problems do hold uniformly in time for the semidiscretization by the same method of the heat equation provided the solution is smooth enough. Thus, if the approximations are piecewise polynomials of degree $k$, the approximation to the gradient converges with the rate $h^{k+1}$ for $k\ge 0$ and the $L^2$-projection of the error into a space of lower polynomial degree superconverges with the rate $\sqrt {\log (T/h^2)} h^{k+2}$ for $k\ge 1$ uniformly in time. As a consequence, an element-by-element postprocessing converges with the rate $\sqrt {\log (T/h^2)} h^{k+2}$ for $k\ge 1$ also uniformly in time. Similar results are proven for the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods.

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations

The discontinuous Galerkin (DG) method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k + 2 superconvergent at Radau points on each element and of order 2 k + 1 superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis.

A Fully-Discrete Local Discontinuous Galerkin Method for Convection-Dominated Sobolev Equation

In this paper we shall present, for the convection-dominated Sobolev equations, the fully-discrete numerical scheme based on the local discontinuous Galerkin (LDG) finite element method and the third-order explicitly total variation diminishing Runge-Kutta (TVDRK3) time marching. A priori error estimate is obtained for any piecewise polynomials of degree at most k?1, under the general spatial-temporal restriction. The bounded constant in error estimate is independent of the reciprocal of the diffusion and dispersion coefficients, after removing the effect of smoothness of the exact solution. Finally some numerical results are given to verify the presented conclusion.

A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows

In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart–Thomas elements of order k ⩾ 0 for the pseudostress, and piecewise polynomials of degree k for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.

Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r?1 for a class of quasi-linear elliptic problems in ???2. We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H 1-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ? d ,d=2,3 and use it to establish the convergence of the two-grid method for problems in ???3.

Polynomial approximation on spheres - generalizing de la Vallée-Poussin

AbstractFor trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}. In all cases the approximation converges uniformly for all continuous functions, and has arbitrarily fast convergence for smooth functions.

On Stability, Accuracy, and Fast Solvers for Finite Element Approximations of the Axisymmetric Stokes Problem by Hood–Taylor Elements

We provide a proof of both the stability and the approximation property for the finite element approximations of the axisymmetric Stokes problem by continuous piecewise polynomials of degree $\kappa+1$ for the velocity and continuous piecewise polynomials of degree $\kappa$ for the pressure with any $\kappa\geq1$. New techniques are designed so that in this perspective, by a simple transformation, the existing theory developed in three dimensional Cartesian coordinates can be effectively exploited. In fact, this perspective provides a new way of developing theories for the axisymmetric Stokes problems and it can be applied potentially to other problems as well. A simple illustration is provided for the application in the development and analysis of fast solvers for the resulting discrete saddle point problems. Sample numerical experiments have been presented as well to confirm the theoretical results.

Bernstein–Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures

Algorithms are presented that enable the element matrices for the standard finite element space, consisting of continuous piecewise polynomials of degree $n$ on simplicial elements in $\mathbb{R}^d$, to be computed in optimal complexity $\mathcal{O}(n^{2d})$. The algorithms (i) take into account numerical quadrature; (ii) are applicable to nonlinear problems; and (iii) do not rely on precomputed arrays containing values of one-dimensional basis functions at quadrature points (although these can be used if desired). The elements are based on Bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements.

The effect of numerical integration on the finite element approximation of linear functionals

In this paper, we have studied the effect of numerical integration on the finite element method based on piecewise polynomials of degree k, in the context of approximating linear functionals, which are also known as “quantities of interest”. We have obtained the optimal order of convergence, $${\mathcal{O}(h^{2k})}$$, of the error in the computed functional, when the integrals in the stiffness matrix and the load vector are computed with a quadrature rule of algebraic precision 2k − 1. However, this result was obtained under an increased regularity assumption on the data, which is more than required to obtain the optimal order of convergence of the energy norm of the error in the finite element solution with quadrature. We have obtained a lower bound of the error in the computed functional for a particular problem, which indicates the necessity of the increased regularity requirement of the data. Numerical experiments have been presented indicating that over-integration may be necessary to accurately approximate the functional, when the data lack the increased regularity.