Piecewise Polynomials Of Degree Research Articles

Negative norm estimates and superconvergence results in Galerkin method for strongly nonlinear parabolic problems

The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree r≥1 are used, which improve upon earlier results of Axelsson ((1977) [3]) requiring for 2d r≥2 and for 3d r≥3. Based on quasi-projection technique introduced by Douglas et al. ((1978) [11]), superconvergence result for the error between Galerkin approximation and approximation through quasi-projection is established for the semidiscrete Galerkin scheme. Further, a priori error estimates in Sobolev spaces of negative index are derived. Moreover, in a single space variable, nodal superconvergence results between the true solution and Galerkin approximation are established.

Ultraconvergence of finite element method by Richardson extrapolation for elliptic problems with inhomogeneous boundary conditions

AbstractIn this article, Richardson extrapolation technique is employed to investigate the local ultraconvergence properties of Lagrange finite element method using piecewise polynomials of degrees () for the second order elliptic problem with inhomogeneous boundary. A sequence of special graded partition are proposed and a new interpolation operator is introduced to achieve order local ultraconvergence for the displacement and derivative.

An approximate solution of integral equation using Bezier control Points

The integral equations are computed numerically using a Bezier curve. We have written the linear Fredholm integral equation into a matrix formulation by using a Bezier curves as a piecewise polynomials of degree n and we use (n+1) unknown control points on unit interval to determine Bezier curve. two examples have been discussed in details.

Adaptive HDG Methods for the Brinkman Equations with Application to Optimal Control

This paper investigates adaptive hybridizable discontinuous Galerkin methods for the gradient-velocity–pressure formulation of Brinkman equations. We use piecewise polynomials of degree k to approximate the velocity, the velocity gradient, the pressure and the boundary traces. By the $$L^2$$ -projection and inf-sup condition, a residual type a posteriori error estimator is introduced for the error measured by an energy norm. Moreover, we prove that the a posteriori error estimator is robust in the sense that the ratio of the upper and lower bounds is independent of the dynamic viscosity and permeability. Then the idea and the result, which have been used and obtained for Brinkman equations, are extended to solve the Brinkman optimal control problem. Based on the variational discretization concept, we also derive an efficient and reliable a posteriori error estimator for the error measured by an energy norm. Finally, several numerical results are provided to validate the theoretical analysis.

Smooth approximation of a varying refractive-index profile and its application in the computation of light waves

In this paper, the smooth approximation of light waves is studied for an open optical waveguide with a distinct refractive-index profile, which involves high-precision computation of the eigenmodes and corresponding eigenfunctions. During analysis, the refractive-index function is first approximated by a quadratic spline interpolation function. Since the quadratic spline function is a polynomial of degree two in every sub-interval (sub-layer), it is equivalent to a piecewise polynomial of degree two, based on which, the corresponding Sturm-Liouville eigenvalue problem of the Helmholtz operator in sub-layer can be solved analytically by the Kummer functions. Finally, the approximate dispersion equation is established to the TE case. Obviously, the approximate dispersion equations converge fast to the exact ones, as the maximum value of the sub-interval sizes tends to zero. Furthermore, eigenmodes may be obtained by Müller’s method with suitable initial values. To refine the accuracy, the equidistant partition and the non-equidistant partition are applied to divide the interval. Numerical simulations show that the eigenfunctions of the spline interpolation are much smoother than the ones with piecewise interpolation. In addition, the non-equidistant partition can help improve the accuracy and the order of convergence of general solutions reaches the third.

Collocation methods for cordial Volterra integro-differential equations

This paper is concerned with collocation methods for cordial Volterra integro-differential equations (CVIDEs) with noncompact cordial operators. The existence, uniqueness and regularity of the exact solutions to CVIDEs are discussed, and a resolvent representation of the derivative of the exact solution is obtained. We approximate the exact solution by collocation in the space of continuous piecewise polynomials of degree m. The solvability of the collocation equations is proved for sufficiently small meshes diameter. It is shown that, if the solution is sufficiently smooth, the collocation solutions are convergent with global order of convergence m. Using an approach based on the resolvent formula, we prove that global superconvergence of order m+1 is attained with iterated collocation based on some special points. Some numerical examples are provided to verify the convergence results.

A fully‐mixed finite element method for the coupling of the Navier–Stokes and Darcy–Forchheimer equations

AbstractIn this work we present and analyze a fully‐mixed formulation for the nonlinear model given by the coupling of the Navier–Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. Our approach yields non‐Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier–Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well‐known Schauder and Banach theorems, combined with classical results on nonlinear monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ≥ 0, Raviart–Thomas spaces of order k, continuous piecewise polynomials of degree ≤k + 1 and piecewise polynomials of degree ≤k are employed in the fluid for approximating the pseudostress tensor, velocity and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ≤k for the velocity and pressure, constitute a feasible choice in the porous medium. A priori error estimates and associated rates of convergence are derived, and several numerical examples illustrating the good performance of the method are reported.

Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions

Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree <i>n</i> joined together at the break points with <i>n</i>-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.

A Generalized Walsh System and Its Fast Algorithm

In this paper, we propose a new class of discontinuous orthogonal system called Walsh-U system, which is composed of piecewise polynomials of degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> in Hilbert space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}[0, 1]$</tex-math></inline-formula> . The Walsh-U system generalizes the Walsh system from piecewise constant to piecewise polynomials, which is capable of representing discontinuous signals. Besides, the basis functions of the Walsh-U system not only possess a more concise construction, but also have stronger sparse representation capabilities than the U-system. In the construction of the Walsh-U system, the Legendre polynomials are adopted as the generators at first, and then a series of basis functions are produced through the generators and basic operations of scaling and replication. These basis functions are formed into different groups based on the generation order. Furthermore, a fast algorithm for the Walsh-U system is designed in this paper. The experimental results demonstrate that when compared with Fourier series, Walsh functions, wavelet functions, and U-system, the Walsh-U system has fast convergence and high approximation precision in analog signal reconstruction and denoising. It is verified that the fast algorithm for the Walsh-U system can effectively improve the computation speed for discontinuous signals.

Asymptotic expansion of iterated Galerkin solution of Fredholm integral equations of the second kind with Green's kernel

We consider a Fredholm integral equation of the second kind with kernel of the type of Green’s function. Iterated Galerkin method is applied to such an integral equation. For r≥1, a space of piecewise polynomials of degree ≤r−1 with respect to a uniform partition is chosen to be the approximating space. We obtain an asymptotic expansion for the iterated Galerkin solution at the partition points. Richardson extrapolation is used to increase the order of convergence. A numerical example is considered to illustrate our theoretical results.

Local ultraconvergence of finite element methods for second‐order elliptic problems with singularity

AbstractIn this article, we will investigate the local ultraconvergence of the displacement and derivative of finite element solution using piecewise polynomials of degrees k (k ≥ 2) by Richardson extrapolation operators and presented in He et al. for second‐order elliptic problems with singularity. Numerical results are provided to illustrate the theoretic findings.

A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations

We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in $$\mathrm {L}^4$$ . As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable $$\mathrm {H}(\mathrm {div})$$ -type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Necas–Babuska theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order $$k\ge 0$$ for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree $$\le k + 1$$ are employed for the velocity, and piecewise polynomials of degree $$\le k$$ for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements) the basis functions are piecewise polynomials of degree $N$ and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree $N$ on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous finite elements on the subtriangles. This choice allows the construction of an efficient, quadrature-free and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efficient Picard iterations. Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time.

An Efficient Solver of Eigenmodes for a Class of Complex Optical Waveguides

In this paper, for a class of complex optical waveguide, the high-precision computation of the propagation constants β are studied. The corresponding Sturm-Liouville (S-L) problem is represented as in an open domain (open on one side), where x is a given value. Firstly, a perfectly matched layer is used to terminate the open domain. Secondly, both the equation and the complex coordinate stretching transformations are constructed. Thirdly, the S-L problem is turned to a simplified form such as in a bounded domain. Finally, the coefficient function is approximated by a piecewise polynomial of degree two. Since the simplified equation in each layer can be solved analytically by the Kummer functions, the approximate dispersion equation is established to the TE case. When the coefficient function is continuous, the approximate solutions converge fast to the exact ones, as the maximum value of the subinterval sizes tends to zero. Numerical simulations show that high-precision eigenmodes may be obtained by the Muller's method with suitable initial values.

A Study of Error Estimation for Second Order Fredholm Integro-Differential Equations

In this work, we study efficient asymptotically correct a posteriori error estimates for the numerical approximation of second order Fredholm integro-differential equations. We use the defect correction principle to find the deviation of the error estimation and show that collocation method by using m degree piecewise polynomial provides order $$\mathcal{O}(h^{m+2})$$ for the deviation of the error. Also, the theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical analysis.

A Banach spaces-based analysis of a new mixed-primal finite element method for a coupled flow-transport problem

In this paper we introduce and analyze a new finite element method for a strongly coupled flow and transport problem in Rn, n∈{2,3}, whose governing equations are given by a scalar nonlinear convection–diffusion equation coupled with the Stokes equations. The variational formulation for this model is obtained by applying a suitable dual-mixed method for the Stokes system and the usual primal procedure for the transport equation. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the solvability analysis, which constitutes the main advantage of the present approach. The resulting continuous and discrete schemes, which involve the Cauchy fluid stress, the velocity of the fluid, and the concentration as the only unknowns, are then equivalently reformulated as fixed point operator equations. Consequently, the well-known Schauder, Banach, and Brouwer theorems, combined with Babuška–Brezzi’s theory in Banach spaces, monotone operator theory, regularity assumptions, and Sobolev imbedding theorems, allow to establish the corresponding well-posedness of them. In particular, Raviart–Thomas approximations of order k≥0 for the stress, discontinuous piecewise polynomials of degree ≤k for the velocity, and continuous piecewise polynomials of degree ≤k+1 for the concentration, becomes a feasible choice for the Galerkin scheme. Next, suitable Strang-type lemmas are employed to derive optimal a priori error estimates. Finally, several numerical results illustrating the performance of the mixed-primal scheme and confirming the theoretical rates of convergence, are provided.

Energy-preserving finite element methods for a class of nonlinear wave equations

In this paper for the first time, two kinds of energy-preserving finite element approximation schemes, which are based upon the standard finite element method (FEM) and the mixed FEM, respectively, are developed and analyzed for a class of nonlinear wave equations. The energy conservation and the optimal convergence properties are obtained for both finite element schemes in their respective norms, additionally, the energy-preserving mixed FEM can produce one-order higher approximation accuracy to the flux (the gradient of the primary unknown) in L2 norm in contrast with that of the standard FEM when the same degree piecewise polynomial is employed to construct their respective finite element spaces, which may likely result in a more accurate and more physical discrete energy conservation. Numerical experiments are carried out to validate all attained theoretical results. Furthermore, the developed energy-preserving finite element methods can be directly applied to the coupled system of nonlinear wave equations, whose energy conservation and optimal convergence properties are also confirmed by our numerical experiments.

EXtended HDG Methods for Second Order Elliptic Interface Problems

In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise $C^2$ smooth interface. It uses piecewise polynomials of degrees $k$ $(k>= 1)$ and $k-1$ respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree $ k$ for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree $ k-1$ for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in $L^2$ norm and for the potential approximation in piecewise $H^1$ seminorm without requiring "sufficiently large" stabilization parameters in the schemes. In addition, error estimation for the potential approximation in $L^2$ norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.

A Robust HDG Method for Reissner-Mindlin Plate Problem

We propose and analyze a hybridizable discontinuous Galerkin (HDG) method for Reissner-Mindlin plate problems. The method uses piecewise-polynomials of degree k(≥ 1) to approximate the transverse displacement and the displacement trace on inter-element boundaries, uses piecewise-polynomial vectors of degrees k and ‘, max(1; k - 1) ≤ ‘ ≤ k, to approximate respectively the rotation and the rotation trace on inter-element boundaries, and uses piecewise-polynomial vectors of degree k and piecewise-polynomial tensors of degree m, k - 1 ≤ m ≤ ‘, to approximate respectively the shear stress and the bending moment. We show that the HDG method is robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the plane thickness t. Numerical experiments are performed to confirm our theoretical results.

Superconvergence results for the nonlinear Fredholm–Hemmerstein integral equations of second kind

The multi-projection methods for solving the Fredholm-Hammerstein integral equation is proposed in this paper. We obtain the similar super-convergence results as in Mandal and Nelakanti (J Comput Appl Math 319:423–439, 2017) with a smooth kernel using piecewise polynomials of degree $$\le r-1,$$ i.e., for both the multi-Galerkin and multi-collocation methods have order of convergence $$\mathcal O (h^{3r})$$ in uniform norm, where h is the norm of the partition. We have also considered iterated version of these methods and prove that both iterated multi-Galerkin and iterated multi-collocation methods have order of convergence $$\mathcal O(h^{4r})$$ in uniform norm. Numerical examples are given to illustrate the theoretical results.