The Experts below are selected from a list of 6000 Experts worldwide ranked by ideXlab platform
Zhimin Zhang - One of the best experts on this subject based on the ideXlab platform.
-
Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives
Journal of Scientific Computing, 2019Co-Authors: Beichuan Deng, Zhimin Zhang, Xuan ZhaoAbstract:In this paper, Superconvergence points are located for the approximation of the Riesz derivative of order $$\alpha $$ using classical Lobatto-type polynomials when $$\alpha \in (0,1)$$ and generalized Jacobi functions (GJF) for arbitrary $$\alpha > 0$$, respectively. For the former, Superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre–Lobatto expansion. It is observed that the convergence rate for different $$\alpha $$ at the Superconvergence points is at least $$O(N^{-2})$$ better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order $$\alpha > 1$$ with the help of GJF, which deal well with the singularities. The well-posedness, convergence and Superconvergence properties are theoretically analyzed. The gain of the convergence rate at the Superconvergence points is analyzed to be $$O(N^{-(\alpha +3)/2})$$ for $$\alpha \in (0,1)$$ and $$O(N^{-2})$$ for $$\alpha > 1$$. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted Superconvergence points.
-
Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations
SIAM Journal on Numerical Analysis, 2018Co-Authors: Waixiang Cao, Yang Yang, Chiwang Shu, Zhimin ZhangAbstract:In this paper, we study the Superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for scalar nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of k+2 and k+1, respectively. As a byproduct, we show a (k+2)th order Superconvergence of the DG solution towards the Gauss--Radau projection of the exact solution. On the other hand, Superconvergence resu...
-
Superconvergence of discontinuous galerkin methods for 1 d linear hyperbolic equations with degenerate variable coefficients
Mathematical Modelling and Numerical Analysis, 2017Co-Authors: Waixiang Cao, Chiwang Shu, Zhimin ZhangAbstract:In this paper, we study the Superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes Superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of O (h k +2 ), when piecewise polynomials of degree k are used. We then prove that the highest Superconvergence rate of the DG solution itself is O (h k +3/2 ) as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain Superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.
-
Superconvergence of immersed finite element methods for interface problems
Advances in Computational Mathematics, 2017Co-Authors: Waixiang Cao, Xu Zhang, Zhimin ZhangAbstract:In this article, we study Superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical Superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired Superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, Superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.
-
Superconvergence Points of Fractional Spectral Interpolation
SIAM Journal on Scientific Computing, 2016Co-Authors: Xuan Zhao, Zhimin ZhangAbstract:We investigate Superconvergence properties of the spectral interpolation involving fractional derivatives. Our interest in this Superconvergence problem is, in fact, twofold: when interpolating function values, we identify the points at which fractional derivatives of the interpolant superconverge; when interpolating fractional derivatives, we locate those points where function values of the interpolant superconverge. For the former case, we apply various Legendre polynomials as basis functions and obtain the Superconvergence points, which naturally unify the Superconvergence points for the first order derivative presented in [Z. Zhang, SIAM J. Numer. Anal., 50 (2012), pp. 2966--2985], depending on orders of the fractional derivatives. While for the latter case, we utilize the Petrov--Galerkin method based on generalized Jacobi functions [S. Chen, J. Shen, and L.-L. Wang, Math. Comp., to appear] and locate the Superconvergence points both for function values and fractional derivatives. Numerical examples ...
Chiwang Shu - One of the best experts on this subject based on the ideXlab platform.
-
analysis of optimal Superconvergence of an ultraweak local discontinuous galerkin method for a time dependent fourth order equation
Mathematical Modelling and Numerical Analysis, 2020Co-Authors: Yong Liu, Qi Tao, Chiwang ShuAbstract:In this paper, we study Superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this Superconvergence is proved to be k + min(3, k ) when piecewise ℙk polynomials with k ≥ 2 are used. We also prove a 2k -th order Superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove Superconvergence of (k + 2)-th and (k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.
-
Superconvergence analysis of the runge kutta discontinuous galerkin methods for a linear hyperbolic equation
Journal of Scientific Computing, 2020Co-Authors: Xiong Meng, Chiwang Shu, Qiang ZhangAbstract:In this paper, we shall establish the Superconvergence property of the Runge–Kutta discontinuous Galerkin (RKDG) method for solving a linear constant-coefficient hyperbolic equation. The RKDG method is made of the discontinuous Galerkin (DG) scheme with upwind-biased numerical fluxes coupled with the explicit Runge–Kutta algorithm of arbitrary orders and stages. Superconvergence results for the numerical flux, cell averages as well as the solution and derivative at some special points are shown, which are based on a systematical study of the $$\hbox {L}^2$$ -norm stability for the RKDG method and the incomplete correction techniques for the well-defined reference functions at each time stage. The result demonstrates that the Superconvergence property of the semi-discrete DG method is preserved, and the optimal order in time is provided under the smoothness assumption that is independent of the number of stages. As a byproduct of the above Superconvergence study, the expected order of the post-processed solution is obtained when a special initial solution is used. Some numerical experiments are also given.
-
Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations
SIAM Journal on Numerical Analysis, 2018Co-Authors: Waixiang Cao, Yang Yang, Chiwang Shu, Zhimin ZhangAbstract:In this paper, we study the Superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for scalar nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of k+2 and k+1, respectively. As a byproduct, we show a (k+2)th order Superconvergence of the DG solution towards the Gauss--Radau projection of the exact solution. On the other hand, Superconvergence resu...
-
Superconvergence of discontinuous galerkin methods for 1 d linear hyperbolic equations with degenerate variable coefficients
Mathematical Modelling and Numerical Analysis, 2017Co-Authors: Waixiang Cao, Chiwang Shu, Zhimin ZhangAbstract:In this paper, we study the Superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes Superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of O (h k +2 ), when piecewise polynomials of degree k are used. We then prove that the highest Superconvergence rate of the DG solution itself is O (h k +3/2 ) as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain Superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.
-
Superconvergence of Discontinuous Galerkin Methods for Two-Dimensional Hyperbolic Equations
SIAM Journal on Numerical Analysis, 2015Co-Authors: Waixiang Cao, Yang Yang, Chiwang Shu, Zhimin ZhangAbstract:This paper is concerned with Superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k+1)th order Superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree k are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of (k+1)th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of (k+2)th order. Numerical experiments indicate that the aforementioned Superconvergence rates are sharp.
Srinivasan Natesan - One of the best experts on this subject based on the ideXlab platform.
-
A unified study on Superconvergence analysis of Galerkin FEM for singularly perturbed systems of multiscale nature
Journal of Applied Mathematics and Computing, 2020Co-Authors: Maneesh Kumar Singh, Gautam Singh, Srinivasan NatesanAbstract:We discuss the Superconvergence analysis of the Galerkin finite element method for the singularly perturbed coupled system of both reaction–diffusion and convection–diffusion types. The Superconvergence study is carried out by using linear finite element, and it is shown to be second-order (up to a logarithmic factor) uniformly convergent in the suitable discrete energy norm. We have conducted some numerical experiments for the system of reaction–diffusion and system of convection–diffusion models, which validate the theoretical results.
Ivo Babuška - One of the best experts on this subject based on the ideXlab platform.
-
Superconvergence in the generalized finite element method
Numerische Mathematik, 2007Co-Authors: Ivo Babuška, Uday Banerjee, John E. OsbornAbstract:In this paper, we address the problem of the existence of Superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the Superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the Superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct.
-
η%-Superconvergence of Finite Element Solutions and Error Estimators
Advances in Computational Mathematics, 2001Co-Authors: Lin Zhang, T. Strouboulis, Ivo BabuškaAbstract:This paper is a summary of our study on the Superconvergence of the finite element solutions and error estimators. We will persent the analysis of η%-Superconvergence for finite element solutions of the Poisson equation in the interior of meshes of triangles with straight edges, as well as the analysis at the boundary. The η%-Superconvergence via local averaging will also be presented, and the error estimators are compared in the sense of η%-Superconvergence.
-
η%-Superconvergence of finite element approximations in the interior of general meshes of triangles
Computer Methods in Applied Mechanics and Engineering, 1995Co-Authors: Ivo Babuška, T. Strouboulis, Chandra Shekhar UpadhyayAbstract:Abstract In this paper we introduce a new definition of Superconvergence — tne η%-Superconvergence, which generalizes the classical idea of Superconvergence to general meshes. We show that this new definition can be employed to determine the regions of least-error in any element in the interior of any grid by using a computer-based approach. We present numerical results for the standard displacement finite element method for the scalar equation of orthotropic heat-conduction, for meshes of conforming triangles of degree p, 1 ⩽ p ⩽ 5, and elements in the interior of the mesh. The results demonstrate that, unlike classical Superconvergence, η%-Superconvergence is applicable to the complex grids which are employed in practical engineering computations.
-
e%-Superconvergence in the interior of locally refined meshes of quadrilaterals: Superconvergence of the gradient in finite element solutions of Laplace's and Poisson's equations
Applied Numerical Mathematics, 1994Co-Authors: Ivo Babuška, T. Strouboulis, S. K. Gangaraj, Chandra Shekhar UpadhyayAbstract:Abstract This paper is the third in a series in which we study the Superconvergence of finite element solutions by a computer-based approach. In [1] we studied classical Superconvergence and in [2] we introduced the new concept of η%-Superconvergence and showed that it can be employed to determine regions of least error for the derivatives of the finite element solution in the interior of any grid of triangular elements. Here we use the same ideas to study the Superconvergence of the derivatives of the finite element solution in the interior of complex grids of quadrilaterals of the type used in practical computations.
-
Study of Superconvergence by a Computer-Based Approach: Superconvergence of the Gradient of the Displacement, The Strain and Stress in Finite Element Solutions for Plane Elasticity.
1994Co-Authors: Ivo Babuška, T. Strouboulis, Chandra Shekhar Upadhyay, S. K. GangarajAbstract:Abstract : In 1 we addressed the problem of existence of Superconvergence points by a computer-based proof and we gave a detailed study of the Superconvergence points for the components of the gradient in finite element solutions for Laplace's and Poisson's equations. Here we employ the same approach to study the Superconvergence for the gradient of the displacement, the strain and the stress for finite element solutions of the equations of plane elasticity. We give the Superconvergence points for the components of the gradient of the displacement, the strain and stress for meshes of triangles and squares of degree p, 1 < or = p < or = 4. For the meshes of triangles we investigated the effect of the topology of the mesh by considering four mesh-patterns which typically occur in practical meshes, while in the case of square elements we studied the effect of the element-type (tensor-product, serendipity or other).
Maneesh Kumar Singh - One of the best experts on this subject based on the ideXlab platform.
-
A unified study on Superconvergence analysis of Galerkin FEM for singularly perturbed systems of multiscale nature
Journal of Applied Mathematics and Computing, 2020Co-Authors: Maneesh Kumar Singh, Gautam Singh, Srinivasan NatesanAbstract:We discuss the Superconvergence analysis of the Galerkin finite element method for the singularly perturbed coupled system of both reaction–diffusion and convection–diffusion types. The Superconvergence study is carried out by using linear finite element, and it is shown to be second-order (up to a logarithmic factor) uniformly convergent in the suitable discrete energy norm. We have conducted some numerical experiments for the system of reaction–diffusion and system of convection–diffusion models, which validate the theoretical results.
