Discontinuous Piecewise Polynomials Research Articles

Influence of Reference-to-Physical Frame Mappings on Approximation Properties of Discontinuous Piecewise Polynomial Spaces

In this manuscript we compare physical and reference frame discontinuous Galerkin (dG) discretizations with emphasis on the influence of reference-to-physical frame mappings on the discrete space properties. We assess the excellence of physical frame discrete spaces in terms of approximation capabilities as well as the increased flexibility compared to reference frame discretizations. As a matter of fact, whenever curved elements are considered, non-affine reference-to-physical frame mappings are able to spoil the convergence properties of reference frame discrete spaces. This poorly documented drawback does not affect basis functions defined directly in the physical frame.

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.

Optimal Error Estimates of the Local Discontinuous Galerkin Method for Surface Diffusion of Graphs on Cartesian Meshes

In (Xu and Shu in J. Sci. Comput. 40:375---390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L 2 norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k?1.

A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations

We are interested in the deterministic computation of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The main difficulty of such computation arises from the very high dimensions of the model, making it necessary to use relatively coarse meshes and hence requiring the numerical solver to be stable and to have good resolution under coarse meshes. In this paper we give a brief survey of the discontinuous Galerkin (DG) method, which is a finite element method using discontinuous piecewise polynomials as basis functions and numerical fluxes based on upwinding for stability, for solving the Boltzmann-Poisson system. In many situations, the deterministic DG solver can produce accurate solutions with equal or less CPU time than the traditional DSMC (Direct Simulation Monte Carlo) solvers. In order to make the presentation more concise and to highlight the main ideas of the algorithm, we use a simplified model to describe the details of the DG method. Sample simulation results on the full Boltzmann-Poisson system are also given.

A class of quaternary orthonormal U-system

In this paper, we present the methods of construction of quaternary orthonormal system (so-called QU-system) with piecewise polynomials, discuss the feasibility of the construction methods, and obtain a set of explicit expressions of QU-system with degrees 1 to 3. And then, we investigate the properties of QU-system and the relationship between binary U-system and QU-system, and present the formulae of its basis value and Fourier-QU coe±cient. Applying the investigated construction methods, we can construct a class of complete orthonormal system in L 2[0,1], which contains both continuous and discontinuous piecewise polynomials. So QU-system has the properties of both Fourier trigonometric functions and Walsh functions. Finally, we apply numerical experiments to confirm that the convergence rate of Fourier-QU series is better than that of Fourier series, Walsh series and Fourier-BU series, if using the first finite terms of them to approximate the functions of one variable.

A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L 2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.

Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics

We study the applicability of the discontinuous Petrov–Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate, local computation of the corresponding ‘optimal’ test functions. In the Timoshenko beam problem, the proposed method is shown to provide the best approximation in an energy-type norm which is equivalent to the L 2-norm for all the unknowns, uniformly with respect to the thickness parameter. The same formulation remains valid also for the asymptotic Euler–Bernoulli solution. As another one-dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations. In particular, we derive a special norm for the test space which leads to a robust method in terms of the shell thickness. Finally, we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp-mesh for resolving the boundary layer.

Mixed Finite Element Methods for Incompressible Flow: Stationary Navier–Stokes Equations

In [Z. Cai, C. Tong, P. S. Vassilevski, and C. Wang, Numer. Methods Partial Differential Equations, to appear], the authors developed and analyzed a mixed finite element method for the stationary Stokes equations based on the pseudostress-velocity formulation. The pseudostress and the velocity are approximated by a stable pair of finite elements: Raviart–Thomas elements of index $k\geq0$ and discontinuous piecewise polynomials of degree $k\geq0$, respectively. This paper extends the method to the stationary, incompressible Navier–Stokes equations. Under appropriate assumptions, we show that the pseudostress-velocity formulation of the Navier–Stokes equation and its discrete counterpart have branches of nonsingular solutions, and error estimates of the mixed finite element approximations are established as well.

Stability of Lagrange elements for the mixed Laplacian

The stability properties of simple element choices for the mixed formulation of the Laplacian are investigated numerically. The element choices studied use vector Lagrange elements, i.e., the space of continuous piecewise polynomial vector fields of degree at most r, for the vector variable, and the divergence of this space, which consists of discontinuous piecewise polynomials of one degree lower, for the scalar variable. For polynomial degrees r equal 2 or 3, this pair of spaces was found to be stable for all mesh families tested. In particular, it is stable on diagonal mesh families, in contrast to its behavior for the Stokes equations. For degree r equal 1, stability holds for some meshes, but not for others. Additionally, convergence was observed precisely for the methods that were observed to be stable. However, it seems that optimal order L 2 estimates for the vector variable, known to hold for r>3, do not hold for lower degrees.

Mollified birth in natural-age-grid Galerkin methods for age-structured biological systems**This paper is published as part of a collection in honour of Todd Dupont's 65th birthday.

We present natural-age-grid Galerkin methods for a model of a biological population undergoing aging. We use a mollified birth term in the method and analysis. The error due to mollification is of arbitrary order, depending on the choice of mollifier.The methods in this paper generalize the methods presented in [1], where the approximation space in age was taken to be a discontinuous piecewise polynomial subspace of L2. We refer to these methods as ‘natural-age-grid’ Galerkin methods since transport in the age variable is computed through the smooth movement of the age grid at the natural dimensionless velocity of one. The time variable has been left continuous to emphasize this smooth motion, as well as the independence of the time and age discretizations. The methods are shown to be superconvergent in the age variable.

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann–Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n + – n – n + diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers.

Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients

In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple postprocessing technique. This method improves an existing approach for these problems and uses Raviart–Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and numerical experiments that exhibit the reduced computational cost of this approach are presented.

Discontinuous Galerkin solver for Boltzmann-Poisson transients

We present results of a discontinuous Galerkin scheme applied to deterministic computations of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n +-n-n + diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation.

Simulation of silicon semiconductor devices by means of a direct Boltzmann‐Poisson solver

Purpose – The purpose of this paper is to present a new deterministic solution method to the coupled Boltzmann‐Poisson system for simulating semiconductor devices.Design/methodology/approach – A non‐parabolic six‐valley model allows for the investigation of anisotropy effects. The solution method is based on a discontinuous piecewise polynomial approximation of the carrier distribution function. Integrating the Boltzmann equation over tiny cells of the phase space leads to a system of ordinary differential equations. The Poisson equation is selfconsistently solved by applying a finite element Galerkin approach.Findings – Good agreement with shock‐capturing “WENO solutions” is obtained for n+‐n‐n+ silicon diodes. The anisotropy due to the six‐valley model affects considerably macroscopic quantities at the beginning of the transients. The method is also applicable to spatially two‐dimensional problems.Research limitations/implications – The presented method is extendable by including full band structure dat...

A new superconvergent collocation method for eigenvalue problems

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r − 1 \leq r-1 , we show that the proposed method exhibits an error of the order of 4 r 4r for eigenvalue approximation and of the order of 3 r 3r for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order 4 r 4r can be obtained. This improves upon the order 2 r 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r r and 2 r 2r for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

We develop a one-parameter family of hp-version discontinuous Galerkin finite element methods, parameterised by θ ∈ [−1, 1], for the numerical solution of quasilinear elliptic equations in divergence form on a bounded open set Ω ⊂ ℝd, d ≥ 2. In particular, we consider the analysis of the family for the equation −∇ ·{μ(x, |∇u|)∇u} = f(x) subject to mixed Dirichlet–Neumann boundary conditions on ∂ Ω. It is assumed that μ is a real-valued function, μ ∈ C(Ω̄ × [0, ∞)), and there exist positive constants mμ and Mμ such that mμ(t − s) ≤ μ(x, t)t − μ(x, s)s ≤ Mμ(t − s) for t ≥ s ≥ 0 and all x ∈ Ω̄. Using a result from the theory of monotone operators for any value of θ ∈ [−1, 1], the corresponding method is shown to have a unique solution uDG in the finite element space. If u ∈ C1(Ω) ∩ Hk(Ω), k ≥ 2, then with discontinuous piecewise polynomials of degree p ≥ 1, the error between u and uDG, measured in the broken H1(Ω)-norm, is 𝒪(hs−1/pk−3/2), where 1 ≤ s ≤ min {p + 1, k}.

The reconstruction of discontinuous piecewise polynomial signals

The Gibbs phenomenon was recognized as early as 1898 by Michelson and Stratton. Gibbs oscillations occur during the reconstruction of discontinuous functions from a truncated periodic series expansion, such as a truncated Fourier series expansion or a truncated discrete Fourier transform expansion. Recent theoretical results have shown that Gibbs oscillations can be removed from the truncated Fourier series representation of a function that has discontinuities. This is accomplished by a change of basis to the set of orthogonal polynomials called the Gegenbauer polynomials. In this correspondence, a straightforward numerical procedure for the denoising of piecewise polynomial signals is developed. Examples using truncated Fourier series and discrete Fourier transform (DFT) series demonstrate the effectiveness of the numerical procedure.

Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincare--Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.

Optimal Superconvergence Orders of Iterated Collocation Solutions for Volterra Integral Equations with Vanishing Delays

In this paper we analyze the optimal convergence properties of collocation approximations to solutions of Volterra integral equations of the second kind with vanishing variable delays. The focus of the analysis is on the superconvergence of the (iterated) collocation approximation corresponding to collocation in the space of (discontinuous) piecewise polynomials of degree $m-1 \geq 0$. We show that on uniform meshes the iterated collocation solution possesses the global superconvergence order $m+1$, and that the solution's order of local superconvergence at the nodes of the underlying mesh cannot exceed $m+2$, in sharp contrast to problems with nonvanishing delays. Moreover, the optimal order $p^{\ast } = m+2$ can be attained only under suitable assumptions. This result also resolves a conjecture of 1997 regarding the attainable order of local superconvergence for Volterra integral equations containing a proportional delay $qt$ with $q \in (0,1)$.

Approximate Solution of Multivariable Integral Equations of the Second Kind

For a multidimensional integral equation of the second kind with a smooth kernel, using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree r, Atkinson has established an order r + 1 convergence for the Galerkin solution and an order 2r+2 convergence for the iterated Galerkin solution. In a recent paper [15], a new method based on projections has been shown to give a 4r + 4 convergence for one-dimensional second kind integral equations. The size of the system of equations that must be solved in implementing this method remains the same as for the Galerkin method. In this paper, this method is extended to multi-dimensional second kind equations and is shown to have convergence of order 4r + 4. For interpolatory projections onto a space of piecewise polynomials, it is shown that the order of convergence of the new method improves on the previously established orders of convergence for the collocation and the iterated collocation methods. A two-grid norm convergent method based on the new method is also defined.