Superconvergence Properties Research Articles

Superconvergence of an Ultra-Weak Discontinuous Galerkin Method for Nonlinear Second-Order Initial-Value Problems

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form [Formula: see text]. Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order [Formula: see text] in the [Formula: see text]-norm for the solution, when piecewise polynomials of degree at most [Formula: see text] are used. We use these results to prove that the UWDG solution is superconvergent with order [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] towards a special projection of the exact solution. We further prove that the [Formula: see text]-degree UWDG solution and its derivative are [Formula: see text] superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree [Formula: see text]. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.

Guaranteed a-posteriori error estimation for semi-discrete solutions of parabolic problems based on elliptic reconstruction

This paper addresses the construction of error estimate for semi-discrete solutions of parabolic problems based on the superconvergence properties in the elliptic reconstruction.

Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems

This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasize that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. In addition, as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.35∘)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84∘)-stable method with nodes symmetric in [0, 1] to a common center that performs equally well. Numerical tests with four well established optimal control problems confirm the theoretical findings also concerning A-stability.

Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

Superconvergence and fast implementation of the barycentric prolate differentiation

First, we study the superconvergence properties of the prolate interpolation and differentiation. Advantages over the polynomial-based results can be observed in approximating and solving differential equation. Then we propose the fast implementation of the second order barycentric prolate differentiation by the fast multipole method (FMM) and the optimal convergence rate is given. Effectiveness and accuracy of the proposed method are tested by numerical examples.

Recovery based finite difference scheme on unstructured mesh

In the article, we propose a systematic method to generate finite difference scheme on unstructured mesh. Our method is based on recovery technique in the finite element community. The computed solutions own the same superconvergence (or ultraconvergence) property as recovered gradient and Hessian.

A new patch up technique for elliptic partial differential equation with irregularities

This paper presents a new technique based on a collocation method using cubic splines for second order elliptic equation with irregularities in one dimension and two dimensions. The differential equation is first collocated at the two smooth sub domains divided by the interface. We extend the sub domains from the interior of the domain and then the scheme at the interface is developed by patching them up. The scheme obtained gives the second order accurate solution at the interface as well as at the regular points. Second order accuracy for the approximations of the first order and the second order derivative of the solution can also be seen from the experiments performed. Numerical experiments for 2D problems also demonstrate the second order accuracy of the present scheme for the solution u and the derivatives ux,uxx and the mixed derivative uxy. The approach to derive the interface relations, established in this paper for elliptic interface problems, can be helpful to derive high order accurate numerical methods. Numerical tests exhibit the super convergent properties of the scheme.

A novel approach of superconvergence analysis of the bilinear-constant scheme for time-dependent Stokes equations

The superconvergent properties for time-dependent Stokes equations are investigated in this paper. Firstly, based on a new approach and the special characters of the bilinear-constant finite element space pair on the rectangular mesh, the superclose error estimations in L∞(H1)-norm for velocity and in L∞(L2)-norm for pressure are obtained. Then, the corresponding superconvergence results for velocity and pressure are derived in terms of a suitable postprocessing approach. Finally, some numerical experiments are provided to verify the theoretical analysis.

A superconvergent hybridizable discontinuous Galerkin method for weakly compressible magnetohydrodynamics

This work proposes a hybridizable discontinuous Galerkin (HDG) method for the solution of magnetohydrodynamic (MHD) problems with weakly compressible flows. A novel fluid formulation that adopts the velocity and the pressure as primal variables is first derived and its superior properties, compared to alternative density–momentum-based approaches, are demonstrated on a simple benchmark. The coupled MHD formulation exhibits superconvergence properties for both the fluid velocity and the magnetic induction, a feature not present in any HDG formulation published in this field. An alternative MHD formulation, adopting a fluid-type solver for the solution of the magnetic subproblem, is also considered and its advantages and disadvantages are discussed. The convergence properties of the proposed formulations for the single physics and for the coupled problem are examined on an extensive set of numerical examples in both two and three dimensions, on structured and unstructured meshes and at low and high Hartmann numbers.

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

AbstractA fully discrete scheme is proposed for multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients on anisotropic meshes, where linear triangular finite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank–Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time–space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.

A superconvergent elastically coupled double beam element for analysis of adhesively bonded lap joints

A novel elastically coupled double beam (ECDB) finite element with superconvergent properties is developed to primarily study wave propagation in the bonded region of an adhesively bonded single lap joint. An ECDB element consists of two axial-flexural-shear coupled beams coupled with continuously distributed linear elastic springs. A general formulation applicable to both metals and composites is proposed. An exact solution to the static part of six-coupled governing equations of motion is obtained, which is then used to formulate exact stiffness and mass matrices. Consequently, the ECDB element thus formulated has superconvergent properties and is inherently free of shear locking. It is shown that merely a single ECDB element is sufficient to accurately capture the tip deflections of a cantilever double beam type geometry subjected to static loading, following which eigenvalue analysis is performed, and comparisons are made with the commercial finite element software—Abaqus. Further, wave propagation studies are carried out to demonstrate the efficiency of the element in evaluating ultrasonic wave responses across various metallic and composite ECDBs, and comparisons are made with Abaqus and frequency-domain spectral finite element (SFEM) model described in Paunikar and Goapalakrishnan (“Wave propagation in adhesively bonded metallic and composite lap joints modelled through spectrally formulated elastically coupled double beam element,” Compos. Struct., under review). Following this, wave propagation in symmetric metallic, geometrically asymmetric metallic, and symmetric laminated composite lap joints with strong and weak bonding is studied by employing two-noded Lagrangian frame elements to mesh the adherends and superconvergent elastically coupled double beam (SECDB) elements to mesh the bonded region. The results obtained from the superconvergent finite element (SCFE) simulations are compared with those given by Abaqus and SFEM. Lastly, the SCFE models for the cases of perfectly bonded aluminium and symmetric ply laminated composite are experimentally validated. We have shown that the SECDB element developed in this work may be used to efficiently carry out static and dynamic analysis in any symmetric or asymmetric bonded joint comprising of only metals or composites or a combination of the two. It is also demonstrated that various levels of adhesion in a bonded joint can be simulated by varying the coupling spring stiffness value. Additionally, the SECDB element may also be used to study other double beam like coupled structures, for instance, space platforms or dynamic vibration absorbers.

Unified analysis for variational time discretizations of higher order and higher regularity applied to non-stiff ODEs

We present a unified analysis for a family of variational time discretization methods, including discontinuous Galerkin methods and continuous Galerkin–Petrov methods, applied to non-stiff initial value problems. Besides the well-definedness of the methods, the global error and superconvergence properties are analyzed under rather weak abstract assumptions which also allow considerations of a wide variety of quadrature formulas. Numerical experiments illustrate and support the theoretical results.

Superconvergence of Discontinuous Galerkin Methods for Elliptic Boundary Value Problems

In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order $$\min (k+1, 2)$$ if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.

How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Higher order accuracy is one of the well-known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. The smoothness-increasing accuracy-conserving (SIAC) filter is a popular post-processing technique introduced by Cockburn et al. (Math. Comput. 72(242): 577–606, 2003). It can raise the convergence rate of the DG solution (with a polynomial of degree k) from order \(k+1\) to order \(2k+1\) in the \(L^2\) norm. This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction. The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features, such as extra smoothness. Second, we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We prove the superconvergence error estimate of the new SIAC filters. Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.

Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations II: HHO-Inspired Methods

In Chen et al. (J. Sci. Comput. 81(3): 2188–2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction-diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree \(k\geqslant 1\). In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3): 635–650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements affine-equivalent to a finite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for \(k\geqslant 0\) by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

Superconvergence analysis of an energy stable scheme with three step backward differential formula‐finite element method for nonlinear reaction–diffusion equation

AbstractA three step backward differential formula scheme is proposed for nonlinear reaction–diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size and the subdivision parameter . Temporal error estimate in H2‐norm is derived, which leads to the boundedness of the solutions of the time‐discrete equations. Unconditional spatial error estimate in L2‐norm is deduced which help bound the numerical solutions in L∞‐norm. Superconvergent property of in H1‐norm with order is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.

Superconvergence error estimate of Galerkin method for Sobolev equation with Burgers' type nonlinearity

In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L∞(H1)-norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

Superconvergence analysis of a MFEM for BBM equation with a stable scheme

In this paper, superconvergence properties are presented for the nonlinear Benjamin-Bona-Mahony (BBM) equation with a mixed finite element method (MFEM). We propose an Euler scheme combined with the artificial Douglas-Dupont regularization terms, which guarantee the stability of the numerical scheme. Splitting technique is utilized to get rid of the ratio between the time step τ and the subdivision parameter h. Temporal error estimates and unconditional spatial error estimates are gained through some techniques, such as the function's monotonicity and the Green formula and so on. In turn, the regularities of the solutions about the time-discrete equations and the boundedness of the numerical solution are derived. Based on the above achievements, the unconditional superconvergent results of un in H1-norm and q→n in L2-norm with order O(h2+τ) are obtained through the trigonometric inequality. The global superconvergent results are deduced by use of the interpolated postprocessing operators. Numerical example shows the validity of the theoretical analysis.

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.

Third order positivity-preserving direct discontinuous Galerkin method with interface correction for chemotaxis Keller-Segel equations

In this article, we apply direct discontinuous Galerkin method with interface correction (DDGIC) to solve chemotaxis Keller-Segel equation and prove the quadratic polynomial solution satisfying positivity-preserving with third order of accuracy. We show DDGIC method can obtain optimal convergence order for the cell density variable without introducing extra auxiliary variables to approximate the gradient of the chemical concentration. This is due to the super convergent property of the DDGIC method. We prove that, with the proper choice of the numerical flux coefficients, the cell density approximation can be preserved none negative at all time. Uniform third order of accuracy is maintained with the positivity-preserving limiter applied. For the chemotaxis model with blow up or singular solutions, the DDGIC method can effectively remove negative values approximations and accurately capture the blow up time.