Order Superconvergence Research Articles

Superconvergent method for weakly singular Fredholm-Hammerstein integral equations with non-smooth solutions and its application

In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results.

Numerical solution of fuzzy stochastic Volterra integral equations with constant delay

This paper considers the nonlinear fuzzy stochastic Volterra integral equations with constant delay, which are general and include many fuzzy stochastic integral and differential equations discussed in literature. Since Doob's martingale inequality is no longer applicable to such equations, a new maximum inequality is obtained. Combining with the Picard approximation method, the existence and uniqueness of solutions to nonlinear fuzzy stochastic Volterra integral equations with constant delay are given. Moreover we prove that the solution behaves stably in the case of small changes of initial values, kernels and nonlinearities. We further develop a Euler-Maruyama (EM) scheme and prove the strong convergence of the scheme. It is shown that the strong convergence order of the EM method is 0.5 under Lipschitz condition. Moreover, the strong superconvergence order is 1 if further, the kernel h(t,s) of the stochastic term satisfies h(t,t)=0. Numerical examples demonstrated that the numerical results are consistent with the theoretical research conclusions. Furthermore, the application model of the fuzzy stochastic Volterra integral equation with constant delay in population dynamics is considered, and the exact solution of the numerical example is given in explicit form.

An explicit Euler scheme for generalized [formula omitted]-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises

The current paper is devoted to the numerical integration of generalized n-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises. We submitted the Euler integrator for the integral form of this type of differential equation. Convergence analysis shows that the proposed scheme order of strong superconvergence is 1.0 when diffusion terms, in general form gl2(t,ρ), l=1,2, satisfies gl2(t,t)=0. While, the strong convergence order of the scheme is 1/2 if gl2(t,t)≠0. Especially, for convolution diffusion terms gl2(t,ρ)=gl2(t−ρ) our scheme has strong superconvergence order 1.0 when gl2(0)=0. And scheme strongly convergent with order 1/2 if gl2(0)≠0. Finally, four problems are considered to exhibit the efficiency and compatibility of the numerical integrator.

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

AbstractIn this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.

A superconvergent CDG finite element for the Poisson equation on polytopal meshes

AbstractA conforming discontinuous Galerkin (CDG) finite element is constructed for solving second order elliptic equations on polygonal and polyhedral meshes. The numerical trace on the edge between two elements is no longer the average of two discontinuous Pk functions on the two sides, but a lifted function from four Pk functions. When the numerical gradient space is the subspace of piecewise polynomials on subtriangles/subtehrahedra of a polygon/polyhedron which have a one‐piece polynomial divergence on , this CDG method has a superconvergence of order two above the optimal order. Due to the superconvergence, we define a post‐process which lifts a Pk CDG solution to a quasi‐optimal solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory.

Explicit two-step peer methods with reused stages

Two-step peer methods for the numerical solution of Initial Value Problems (IVP) combine the advantages of Runge-Kutta (RK) and multistep methods to obtain high stage order and provide in a natural way a dense output. In general, explicit s-stage peer methods require s evaluations of the vector field at each step. Nevertheless, Klinge and coworkers (BIT Numer Math, 2018) have shown that some methods use less function calls se<s, here called effective stages, by re-using sr=s−se previously computed stages (shifted stages) from the previous steps in the current one.In this paper we propose a new approach, different from the one used by Klinge and coworkers, to re-use previously computed stages, that we call peer methods with reused stages, showing that methods with reused stages and se effective stages are equivalent to three-step peer methods with se stages. Then, we analyze all the families of methods with two effective stages, obtaining methods with s=3 and orders 4 and 5 in which the free parameters of the families have been used to minimize the coefficient of the leading error term as well as to maximize the absolute stability interval. We have also studied one family of peer methods with s=4 and three effective stages, obtaining a method with order 6, superconvergent of order 7, and optimized leading error term as well as absolute stability interval. Some numerical experiments show the performance of the obtained methods by comparing them with other previously obtained peer methods as well as other standard Runge-Kutta and multistep methods.

Unconditional superconvergence analysis of nonconforming [formula omitted] finite element method for the nonlinear coupled predator-prey equations

A nonconforming EQ1rot finite element method (FEM) is studied for the nonlinear coupled predator-prey equations. The superconvergence estimates are derived for the semi-discrete and Crank-Nicolson (C-N) fully discrete schemes based on the two special properties of this element: one is that the interpolation operator is equivalent to its projection operator, and the other is that the consistency error can be estimated as order O(h2) in the broken H1-norm when the exact solution belongs to H3(Ω), which is one order higher than that of its interpolation error estimate. Moreover, the unique solvability of the nonlinear coupled semi-discrete scheme is certified through the Brouwer fixed point theorem analytically. On the other hand, the stability of the decoupled C-N fully discrete scheme is proved by mathematical induction, which leads to the unconditional superconvergence of order O(h2+τ2) without the ratio between the time-step τ and the mesh size h. Finally, numerical examples are given to demonstrate the validity of our method.

On the comparative performance of fourth order Runge-Kutta and the Galerkin-Petrov time discretization methods for solving nonlinear ordinary differential equations with application to some mathematical models in epidemiology

&lt;abstract&gt;&lt;p&gt;Anti-viral medication is comparably incredibly beneficial for individuals who are infected with numerous viruses. Mathematical modeling is crucial for comprehending the various relationships involving viruses, immune responses and health in general. This study concerns the implementation of a &lt;italic&gt;continuous&lt;/italic&gt; Galerkin-Petrov time discretization scheme with mathematical models that consist of nonlinear ordinary differential equations for the hepatitis B virus, the Chen system and HIV infection. For the Galerkin scheme, we have two unknowns on each time interval which have to be computed by solving a $ 2 \times 2 $ block system. The proposed method is accurate to order 3 in the whole time interval and shows even super convergence of order 4 in the discrete time points. The study presents the accurate solutions achieved by means of the aforementioned schemes, presented numerically and graphically. Further, we implemented the classical fourth-order Runge-Kutta scheme accurately and performed various numerical tests for assessing the efficiency and computational cost (in terms of time) of the suggested schemes. The performances of the fourth order Runge-Kutta and the Galerkin-Petrov time discretization approaches for solving nonlinear ordinary differential equations were compared, with applications towards certain mathematical models in epidemiology. Several simulations were carried out with varying time step sizes, and the efficiency of the Galerkin and Runge Kutta schemes was evaluated at various time points. A detailed analysis of the outcomes obtained by the Galerkin scheme and the Runge-Kutta technique indicates that the results presented are in excellent agreement with each other despite having distinct computational costs in terms of time. It is observed that the Galerkin scheme is noticeably slower and requires more time in comparison to the Runge Kutta scheme. The numerical computations demonstrate that the Galerkin scheme provides highly precise solutions at relatively large time step sizes as compared to the Runge-Kutta scheme.&lt;/p&gt;&lt;/abstract&gt;

DIFFERENCE QUOTIENT ESTIMATES AND ACCURACY ENHANCEMENT OF DISCONTINUOUS GALERKIN METHODS FOR NONLINEAR CONVECTION-DIFFUSION EQUATIONS

In this paper, we apply the post-processing technique to the improvement of the superconvergence of the discontinuous Galerkin method for the nonlinear convection-diffusion equations. We firstly analyze the error estimate and convergence accuracy under <i>L</i>²-norm, and then demonstrate that the α-order difference quotient of DG error is of order <i>k</i>+3/2-<i>α</i>/2 when the upwind fluxes are used. By the duality argument, we construct an appropriate dual equation, and futher obtain superconvergence results of order in the negative-order norm, namely 2<i>k</i>+3/2-<i>α</i>/2 order superconvergence accuracy. Finally, we choose an appropriate kernel function and apply the SIAC filter to the nonlinear convection-diffusion equation to obtain at least 3<i>k</i>/2+1 order superconvergence for post-processed solutions. All theoretical results are proved by numerical experiments.

Characteristic methods for thermal convection problems with infinite Prandtl number on nonuniform staggered grids

In this paper, two characteristic methods are introduced to solve the thermal convection problems with infinite Prandtl number on nonuniform staggered grids. We obtain spatial second order superconvergence in the discrete L2 norm for the temperature, velocity and pressure. Besides, second order superconvergence in the discrete H1 norm for both the temperature and some terms of the velocity on nonuniform grids is established. First and second order time discretizations of the thermal balance equation are provided. Numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

We propose and analyze a modified weak Galerkin finite element method (MWG-FEM) for solving singularly perturbed problems of convection-dominated type. The proposed method is constructed over piecewise polynomials of degree $k\geq1$ on interior of each element and piecewise constant on the boundary of each element. The present method is parameter-free and has less degrees of freedom compared to the classical weak Galerkin finite element method. The method is shown uniformly convergent for small perturbation parameters. An uniform convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy-like norm is established on the piecewise uniform Shishkin mesh, where $N$ is the number of elements. Various numerical examples are presented to confirm the theoretical results. Moreover, we numerically confirm that the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2$-norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L^\infty$-norm.

Dynamical behaviour of HIV Infection with the influence of variable source term through Galerkin method

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD4+ T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host’s blood is considered a separate class. It is reported that when HIV enters a human’s body, it infects a large amount of CD4+ T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD4+ T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2×2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems.

A PRELIMINARY STUDY ON ACCURACY IMPROVEMENT OF NODAL DISPLACEMENTS IN 2D FINITE ELEMENT METHOD

The nodal displacements gain higher convergence rate compared with displacements elsewhere in Finite Element Method (FEM). When the problem is smooth enough, the errors of displacements at element corner nodes can gain convergence order of at most \begin{document}$2m$\end{document} using elements of degree \begin{document}$m$\end{document} in 2D FEM. In this paper, taking the 2D Poisson equation as the model problem and based on an obtained FEM solution, residual nodal load vectors are derived by using super-convergent solutions calculated by Element Energy Projection (EEP) technique. Without changing global stiffness matrices, simple back-substitutions alone would yield nodal displacements of higher convergence rate. Numerical examples show that the nodal displacements can gain super-convergence order of \begin{document}$2m + 2$\end{document} at most when EEP simplified form is used to calculate the residual load vectors. In particular, for linear elements, the accuracy is doubled, and the benefit is very significant.

Superconvergence of the space-time discontinuous Galerkin method for linear nonhomogeneous hyperbolic equations

In this study, we discuss the superconvergence of the space-time discontinuous Galerkin method for the first-order linear nonhomogeneous hyperbolic equation. By using the local differential projection method to construct comparison function, we prove that the numerical solution is $$(2n+1)$$ -th order superconvergent at the downwind-biased Radau points in the discrete $$L^2$$ -norm. As a by-product, we obtain a point-wise superconvergence with order $$2n+\frac{1}{2}$$ in vertices. We also find that, in order to obtain these superconvergence results, the source integral term has to be approximated by $$(n+1)$$ -point Radau-quadrature rule. Numerical results are presented to verify our theoretical findings.

Superconvergence of the Local Discontinuous Galerkin Method for One Dimensional Nonlinear Convection-Diffusion Equations

In this paper, we study superconvergence properties of the local discontinuous Galerkin (LDG) methods for solving nonlinear convection-diffusion equations in one space dimension. The main technicality is an elaborate estimate to terms involving projection errors. By introducing a new projection and constructing some correction functions, we prove the $$(2k+1)$$ th order superconvergence for the cell averages and the numerical flux in the discrete $$L^2$$ norm with polynomials of degree $$k\ge 1$$ , no matter whether the flow direction $$f'(u)$$ changes or not. Superconvergence of order $$k +2$$ ( $$k +1$$ ) is obtained for the LDG error (its derivative) at interior right (left) Radau points, and the convergence order for the error derivative at Radau points can be improved to $$k+2$$ when the direction of the flow doesn’t change. Finally, a supercloseness result of order $$k+2$$ towards a special Gauss–Radau projection of the exact solution is shown. The superconvergence analysis can be extended to the generalized numerical fluxes and the mixed boundary conditions. All theoretical findings are confirmed by numerical experiments.

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 multipoint stress mixed finite element method for elasticity on quadrilateral grids

AbstractWe develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method utilizes the lowest order Brezzi–Douglas–Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell‐centered system for displacement and rotation. The second uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell‐centered system for the displacement only. Stability and error analysis is performed for both methods. First‐order convergence is established for all variables in their natural norms. A duality argument is employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory.

Analysis of optimal superconvergence of an ultraweak-local discontinuous Galerkin method for a time dependent fourth-order equation

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.

Modified weak Galerkin method with weakly imposed boundary condition for convection-dominated diffusion equations

In this paper, a modified weak Galerkin (MWG) finite element method with weakly imposed boundary conditions is presented for solving convection-dominated diffusion equations. The method is shown uniformly stable for all diffusion parameters. The method converges at the optimal order for large diffusion problems in the energy norm, and at half a super-convergent order for small diffusion problems. Various numerical examples are presented, showing that the method is as effective as the weak Galerkin method.

A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems

In this paper, an anisotropic bilinear finite element method is constructed for the elliptic boundary layer optimal control problems. Supercloseness properties of the numerical state and numerical adjoint state in a \begin{document}$ \epsilon $\end{document} -norm are established on anisotropic meshes. Moreover, an interpolation type post-processed solution is shown to be superconvergent of order \begin{document}$ O(N^{-2}) $\end{document} , where the total number of nodes is of \begin{document}$ O(N^2) $\end{document} . Finally, numerical results are provided to verify the theoretical analysis.