Degree Of Polynomial Approximation Research Articles

On the finite element method for a nonlocal degenerate parabolic problem

The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of degree k≥1. Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.

Adaptive scattered data fitting by extension of local approximations to hierarchical splines

We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of (variable degree) polynomial approximations according not only to the number of data points locally available, but also to the smallest singular value of the local collocation matrices. These local approximations are subsequently combined without the need of additional computations with the construction of hierarchical quasi-interpolants described in terms of truncated hierarchical B-splines. A selection of numerical experiments shows the effectivity of our approach for the approximation of real scattered data sets describing different terrain configurations.

Anisotropic [formula omitted]-mesh optimization technique based on the continuous mesh and error models

We develop a new mesh adaptive technique for the numerical solution of partial differential equations (PDEs) using the hp-version of the finite element method (hp-FEM). The technique uses a combination of approximation and interpolation error estimates to generate anisotropic triangular elements as well as appropriate polynomial approximation degrees. We present a hp-version of the continuous mesh model as well as the continuous error model which are used for the formulation of a mesh optimization problem. Solving the optimization problem leads to hp-mesh with the smallest number of degrees of freedom, under the constraint that the approximate solution has an error estimate below a given tolerance. Further, we propose an iterative algorithm to find a suitable anisotropic hp-mesh in the sense of the mesh optimization problem. Several numerical examples demonstrating the efficiency and applicability of the new method are presented.

ERRORS IN FITTING RADIAL DOSE FUNCTION OF COBALT SOURCES FOR BRACHYTHERAPY WITH 3-5 DEGREE POLYNOMIALS

Background: Providing quality insurance for radiation therapy implies high level of precision for determining absorbed dose, and therefore high level of precision for fitting radial dose function. 3-5 degree polynomials provide required precision, however, uncertainty of their coefficients may cause substantial errors. Aim: To investigate dependence of errors of calculating radial dose function with consideration of uncertainty of coefficients from different degrees of fitting polynomial. Materials and methods: Calculations were performed with software package GEANT4.9.6. Geometry and materials of the source correspond to the model BEBIGCo0.A86. Spectral structure of the source corresponds to the NuDat 2.6 database. Statistical processing was done using nonlinear least-square method. Results: Values of the radial dose function of Cobalt source for brachytherapy were calculated for given geometry. Conducted comparison of precisions of 3 to 5 degree polynomial approximations and possible uncertainties of results of radial dose function calculations. Conclusion: Withrequired precision of 25% and higher at the radius of 10 cm the optimal choice for radial dose approximation is the 3rd degree polynomial.

Solving OLG Models with Many Cohorts, Asset Choice and Large Shocks

The paper presents a computationally efficient method to solve overlapping generations models with asset choice. The method is used to study an OLG economy with many cohorts, up to 3 different assets, stochastic volatility, short-sale constraints, and subject to rather large technology shocks. On the methodological side, the main findings are that global projection methods with polynomial approximations of degree 3 are sufficient to provide a very precise solution, even in the case of large shocks. Globally linear approximations, in contrast to local linear approximations, are sufficient to capture the most important financial statistics, including not only the average risk premium, but also the variation of the risk premium over the cycle. However, global linear approximations are not sufficient to reliably pin down asset choices. With a risk aversion parameter of only 4, the model generates a price of risk, measured as the Sharpe ratio, that is almost half of what it is for US stocks. However, the asset price fluctuations and the equity premium are much smaller than in US data.

Anisotropic hp-adaptive method based on interpolation error estimates in the H 1-seminorm

We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken H 1-seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of the proposed anisotropic adaptive strategy in comparison with other adaptive approaches.

Savitzky–Golay filter for reactivity calculation

A new filter method known as Savitzky–Golay allows the reduction of reactivity fluctuations. The filter reduces fluctuations that are found in the nuclear power signal does not attenuate to the reactivity value. The method can be applied with a time step of up to T = 0.01 s and a noise level of up to σ = 0.1. This formulation employs a Gram polynomial approximation of degree d = 2, to calculate the convolution coefficients by means of an analytic formula that is implemented computationally and avoids ill-conditioning issues caused by the inversion of a linear system. The results show better values in the maximum difference and the mean absolute errors of reactivity in comparison with the results reported in the literature.

On the development of an implicit high-order Discontinuous Galerkin method for DNS and implicit LES of turbulent flows

In recent years Discontinuous Galerkin (DG) methods have emerged as one of the most promising high-order discretization techniques for CFD. DG methods have been successfully applied to the simulation of turbulent flows by solving the Reynolds averaged Navier–Stokes (RANS) equations with first-moment closures. More recently, due to their favorable dispersion and dissipation properties, DG discretizations have also been found very well suited for the Direct Numerical Simulation (DNS) and Implicit Large Eddy Simulation (ILES) of turbulent flows.The growing interest in the implementation of DG methods for DNS and ILES is motivated by their attractive features. In particular, these methods can easily achieve high-order accuracy on arbitrarily shaped elements and are perfectly suited to hp-adaptation techniques. Moreover, their compact stencil is independent of the degree of polynomial approximation and is thus well suited for implicit time discretization and for massively parallel implementations.In this paper we focus on recent developments and applications of an implicit high-order DG method for the DNS and ILES of both compressible and incompressible flows. High-order spatial and temporal accuracy has been achieved using the same numerical technology in both cases. Numerical inviscid flux formulations are based on the exact solution of Riemann problems (suitably perturbed in the incompressible case), and viscous flux discretizations rely on the BR2 scheme. Several types of high-order (up to order six) implicit schemes, suited also for DAEs, can be employed for accurate time integration. In particular, linearly implicit Rosenbrock-type Runge–Kutta schemes have been used for all the simulations presented in this work.The massively separated incompressible flow past a sphere at ReD=1000, with transition to turbulence in the wake region, is considered as a DNS test case, while the potential of the ILES is demonstrated by computing the compressible transitional flow at Rec=60 000, M∞=0.1 and α=8∘, around the Selig–Donovan 7003 airfoil. The computed solutions are compared with experimental data and numerical results available in the literature, showing good agreement.

On the stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of the nonstationary nonlinear convection-diffusion initial-boundary value problem in a time-dependent domain formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The nonlinear convection terms are discretized with the aid of a numerical flux. The space discretization uses piecewise polynomial approximations of degree not greater than p with an integer p ⩾ 1. In the theoretical analysis, the piecewise linear time discretization is used. The main attention is paid to the investigation of unconditional stability of the method.

Residual based error estimates for the space–time discontinuous Galerkin method applied to the compressible flows

We develop an adaptive numerical method for solution of the non-stationary compressible Navier–Stokes equations. This method is based on the space–time discontinuous Galerkin discretization, which employs high polynomial approximation degrees with respect to the space as well as to the time coordinates. We focus on the identification of the computational errors, following from the space and time discretizations and from the inexact solution of the arising nonlinear algebraic systems. We derive the residual-based error estimates approximating these errors. Then we propose an efficient algorithm which brings the algebraic, spatial and temporal errors under control. The computational performance of the proposed method is demonstrated by numerical experiments.

Non-uniform spline recovery from small degree polynomial approximation

We investigate the sparse spikes deconvolution problem onto spaces of algebraic polynomials. Our framework encompasses the measure reconstruction problem from a combination of noiseless and noisy moment measurements. We study a TV-norm regularization procedure to localize the support and estimate the weights of a target discrete measure in this frame. Furthermore, we derive quantitative bounds on the support recovery and the amplitude errors under a Chebyshev-type minimal separation condition on its support. Incidentally, we study the localization of the knots of non-uniform splines when a Gaussian perturbation of their inner-products with a known polynomial basis is observed (i.e. a small degree polynomial approximation is known) and the boundary conditions are known. We prove that the knots can be recovered in a grid-free manner using semidefinite programming.

Anisotropic hp-adaptive discontinuous Galerkin method for the numerical solution of time dependent PDEs

We deal with the numerical solution of time dependent problems with the aid of anisotropic hp-grids. We present an algorithm which generates a sequence of anisotropic triangular grids and the corresponding polynomial approximation degrees in such a way that the interpolation error measured in the discrete L∞(0,T;Lq(Ω))-norm (q∈[1,∞] and Ω⊂R2) is under a given tolerance and the number of degrees of freedom is as small as possible. The efficiency of the algorithm is demonstrated by numerical experiments.

Multigrid Algorithms for $hp$-Discontinuous Galerkin Discretizations of Elliptic Problems

We present W-cycle $h$-, $p$-, and $hp$-multigrid algorithms for the solution of the linear system of equations arising from a wide class of $hp$-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in geometric multigrid analysis, we define a smoothing and an approximation property, which are used to prove uniform convergence of the W-cycle scheme with respect to the discretization parameters and the number of levels, provided the number of smoothing steps is chosen of order $p^{2+\mu}$, where $p$ is the polynomial approximation degree and $\mu=0,1$. A discussion on the effects of employing inherited or noninherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results.

A high‐order discontinuous Galerkin method for all‐speed flows

SUMMARYIn this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k − ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.

Anisotropic hp-adaptive method based on interpolation error estimates in the [formula omitted]-norm

We present a new anisotropic hp-adaptive technique, which can be employed for the numerical solution of various scientific and engineering problems governed by partial differential equations in 2D with the aid of a discontinuous piecewise polynomial approximation. This method generates anisotropic triangular grids and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the Lq-norm (q∈[1,∞]). We develop the theoretical background of this approach and present several numerical examples demonstrating the efficiency of the anisotropic adaptive strategy.

Spectral Analysis and Spectral Symbol of $d$-variate $\mathbb Q_{\boldsymbol p}$ Lagrangian FEM Stiffness Matrices

We study the spectral properties of the stiffness matrices coming from the approximation of a $d$-dimensional second order elliptic differential problem by the $\mathbb Q_{\boldsymbol p}$ Lagrangian finite element method (FEM); here, ${\boldsymbol p}=(p_1,\ldots,p_d)$ and $p_j$ represents the polynomial approximation degree in the $j$th direction. After presenting a construction of these matrices, we investigate the conditioning and the spectral distribution in the Weyl sense, and we determine the so-called (spectral) symbol describing the asymptotic spectrum. We also study the properties of the symbol, which turns out to be a $d$-variate function taking values in the space of $N({\boldsymbol p})\times N({\boldsymbol p})$ Hermitian matrices, where $N({\boldsymbol p})=\prod_{j=1}^d p_j$. Unlike the stiffness matrices coming from the ${\boldsymbol p}$-degree B-spline isogeometric analysis approximation of the same differential problem, where a unique $d$-variate real-valued function describes all the spectrum, here the spectrum is described by $N({\boldsymbol p})$ different functions, i.e., the $N({\boldsymbol p})$ eigenvalues of the symbol, which are well-separated, far away, and exponentially diverging with respect to ${\boldsymbol p}$ and $d$. This very involved picture provides an explanation of: (a) the difficulties encountered in designing robust solvers, with convergence speed independent of the matrix size, of the approximation parameters ${\boldsymbol p}$, and of the dimensionality $d$; (b) the convergence deterioration of known iterative methods, already observed in practice for moderate ${\boldsymbol p}$ and $d$. (The PDF has been changed.)

Higher order uniformly convergent continuous/discontinuous Galerkin methods for singularly perturbed problems of convection-diffusion type

In this paper, we propose and analyze a higher order continuous/discontinuous Galerkin methods for solving singularly perturbed convection-diffusion problems. Based on piecewise polynomial approximations of degree k⩾1, a uniform convergence rate O(N−klnkN) in associated norm is established on Shishkin mesh, where N is the number of elements. Numerical experiments complement the theoretical results.

A high‐order accurate discontinuous Galerkin finite element method for laminar low Mach number flows

SUMMARYIn this paper we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of laminar flow simulations at low Mach numbers using an implicit scheme. The algorithm is based on the flux preconditioning approach, which modifies only the dissipative terms of the numerical flux. This formulation is quite simple to implement in existing implicit DG codes, it overcomes the time‐stepping restrictions of explicit multistage algorithms, is consistent in time and thus applicable to unsteady flows. The performance of the method is demonstrated by solving the flow around a NACA0012 airfoil and on a flat plate, at different low Mach numbers using various degrees of polynomial approximations. Computations with and without flux preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers. The time accurate solution of unsteady flow is also demonstrated by solving the vortex shedding behind a circular cylinder at the Reynolds number of 100. Copyright © 2012 John Wiley & Sons, Ltd.

An application of Discontinuous Galerkin space and velocity discretisations to the solution of a model kinetic equation

An approach based on a Discontinuous Galerkin discretisation is proposed for the Bhatnagar–Gross–Krook model kinetic equation. This approach allows for a high-order polynomial approximation of molecular velocity distribution function both in spatial and velocity variables. It is applied to model one-dimensional normal shock wave and heat transfer problems. Convergence of solutions with respect to the number of spatial cells and velocity bins is studied, with the degree of polynomial approximation ranging from zero to four in the physical space variable and from zero to eight in the velocity variable. This approach is found to conserve mass, momentum and energy when high-degree polynomial approximations are used in the velocity space. For the shock wave problem, the solution is shown to exhibit accelerated convergence with respect to the velocity variable. Convergence with respect to the spatial variable is in agreement with the order of the polynomial approximation used. For the heat transfer problem, it was observed that convergence of solutions obtained by high-degree polynomial approximations is only second order with respect to the resolution in the spatial variable. This is attributed to the temperature jump at the wall in the solutions. The shock wave and heat transfer solutions are in excellent agreement with the solutions obtained by a conservative finite volume scheme.

Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains

We present a technique for numerically solving Dirichlet boundary-value problems for second-order elliptic equations on domains $\Omega$ with curved boundaries. This is achieved by using suitably defined extensions from polyhedral subdomains $\textsf{D}_h$; the problem of dealing with curved boundaries is thus reduced to the evaluations of simple line integrals. The technique is independent of the representation of the boundary, of the space dimension, and allows for the use of only polyhedral elements. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that the convergence properties of the resulting method are the same as those for the case in which $\Omega=\textsf{D}_h$ whenever the distance of $\textsf{D}_h$ to $\partial\Omega$ is of order $h$. In particular, we find that when using polynomial approximations of degree $k$ on the polyhedral subdomains $\textsf{D}_h$, both the scalar and vector unknowns converge with the optimal order of $k+1$ in the whole domain $\Omega$ for $k\ge0$. Moreover, we also find that for $k\ge1$, both a postprocessing of the scalar unknown in the polyhedral subdomain $\textsf{D}_h$ as well as its extension from it converge with order $k+2$. The extension also converges with order $2$ for $k=0$.