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Post-processing techniques are essential tools for enhancing the accuracy of finite element approximations and achieving superconvergence. Among these, recovery techniques stand out as vital methods, playing significant roles in both post-processing and pre-processing. This paper provides an overview of recent developments in recovery techniques and their applications in adaptive computations. The discussion encompasses both gradient recovery and Hessian recovery methods. To establish the superconvergence properties of these techniques, two theoretical frameworks are introduced. Applications of these methods are demonstrated in constructing asymptotically exact {\it a posteriori} error estimators for second-order elliptic equations, fourth-order elliptic equations, and interface problems. Numerical experiments are performed to evaluate the asymptotic exactness of recovery type a posteriori error estimators., Comment: To appear in Advances in Applied Mechanics

The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical methods. To address this, we propose a novel class of mapped Hermite functions, which are constructed by applying a mapping to Hermite polynomials.We establish fundamental approximation theory for the orthogonal functions. We propose MHFs-spectral collocation method and MHFs-smoothing transformation method to solve the two-point weakly singular FHIEs, respectively. Error analysis and numerical results demonstrate that our methods, based on the new orthogonal functions, are particularly effective for handling problems with weak singularities at two endpoints, yielding exponential convergence rate. We position this work as the first to directly study the mapped spectral method for multi-point singularity problems, to the best of our knowledge.

Monge-Amp\`{e}re equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the $C^0$ interior penalty method to approximation the viscosity solution of the Monge-Amp\`{e}re equation. The new methods is inspired from the discrete Miranda-Talenti estimate. Based on the vanishing moment representation, we approximate the Monge-Amp\`{e}re equation by the fourth order semi-linear equation with some additional boundary conditions. We will use the discrete Miranda-Talenti estimates to ensure the well-posedness of the numerical scheme and derive the error estimates.

In this paper, we propose and analyze both conforming and nonconforming virtual element methods (VEMs) for the fully nonlinear second-order elliptic Hamilton-Jacobi-Bellman (HJB) equations with Cordes coefficients. By incorporating stabilization terms, we establish the well-posedness of the proposed methods, thus avoiding the need to construct a discrete Miranda-Talenti estimate. We derive the optimal error estimate in the discrete $H^2$ norm for both numerical formulations. Furthermore, a semismooth Newton's method is employed to linearize the discrete problems. Several numerical experiments using the lowest-order VEMs are provided to demonstrate the efficacy of the proposed methods and to validate our theoretical results.

This paper presents an innovative continuous linear finite element approach to effectively solve biharmonic problems on surfaces. The key idea behind this method lies in the strategic utilization of a surface gradient recovery operator to compute the second-order surface derivative of a piecewise continuous linear function defined on the approximate surface, as conventional notions of second-order derivatives are not directly applicable in this context. By incorporating appropriate stabilizations, we rigorously establish the stability of the proposed formulation. Despite the presence of geometric error, we provide optimal error estimates in both the energy norm and $L^2$ norm. Theoretical results are supported by numerical experiments.

Solutions exhibiting weak initial singularities arise in various equations, including diffusion and subdiffusion equations. When employing the well-known L1 scheme to solve subdiffusion equations with weak singularities, numerical simulations reveal that this scheme exhibits varying convergence rates for different choices of model parameters (i.e., domain size, final time $T$, and reaction coefficient $\kappa$). This elusive phenomenon is not unique to the L1 scheme but is also observed in other numerical methods for reaction-diffusion equations such as the backward Euler (IE) scheme, Crank-Nicolson (C-N) scheme, and two-step backward differentiation formula (BDF2) scheme. The existing literature lacks an explanation for the existence of two different convergence regimes, which has puzzled us for a long while and motivated us to study this inconsistency between the standard convergence theory and numerical experiences. In this paper, we provide a general methodology to systematically obtain error estimates that incorporate the exponential decaying feature of the solution. We term this novel error estimate the `decay-preserving error estimate' and apply it to the aforementioned IE, C-N, and BDF2 schemes. Our decay-preserving error estimate consists of a low-order term with an exponential coefficient and a high-order term with an algebraic coefficient, both of which depend on the model parameters. Our estimates reveal that the varying convergence rates are caused by a trade-off between these two components in different model parameter regimes. By considering the model parameters, we capture different states of the convergence rate that traditional error estimates fail to explain. This approach retains more properties of the continuous solution. We validate our analysis with numerical results., Comment: 25 pages, 4 figures

This paper focuses on the superconvergence analysis of the Hessian recovery technique for the $C^0$ Interior Penalty Method (C0IP) in solving the biharmonic equation. We establish interior error estimates for C0IP method that serve as the superconvergent analysis tool. Using the argument of superconvergence by difference quotient, we prove superconvergent results of the recovered Hessian matrix on translation-invariant meshes. The Hessian recovery technique enables us to construct an asymptotically exact ${\it a\, posteriori}$ error estimator for the C0IP method. Numerical experiments are provided to support our theoretical results.

The nonlocality of the fractional operator causes numerical difficulties for long time computation of the time-fractional evolution equations. This paper develops a high-order fast time-stepping discontinuous Galerkin finite element method for the time-fractional diffusion equations, which saves storage and computational time. The optimal error estimate $O(N^{-p-1} + h^{m+1} + \varepsilon N^{r\alpha})$ of the current time-stepping discontinuous Galerkin method is rigorous proved, where $N$ denotes the number of time intervals, $p$ is the degree of polynomial approximation on each time subinterval, $h$ is the maximum space step, $r\ge1$, $m$ is the order of finite element space, and $\varepsilon>0$ can be arbitrarily small. Numerical simulations verify the theoretical analysis., Comment: 21 pages, 1 figure,4 tables

Numerous error estimates have been carried out on various numerical schemes for subdiffusion equations. Unfortunately most error bounds suffer from a factor $1/(1-\alpha)$ or $\Gamma(1-\alpha)$, which blows up as the fractional order $\alpha\to 1^-$, a phenomenon not consistent with regularity of the continuous problem and numerical simulations in practice. Although efforts have been made to avoid the factor blow-up phenomenon, a robust analysis of error estimates still remains incomplete for numerical schemes with general nonuniform time steps. In this paper, we will consider the $\alpha$-robust error analysis of convolution-type schemes for subdiffusion equations with general nonuniform time-steps, and provide explicit factors in error bounds with dependence information on $\alpha$ and temporal mesh sizes. As illustration, we apply our abstract framework to two widely used schemes, i.e., the L1 scheme and Alikhanov's scheme. Our rigorous proofs reveal that the stability and convergence of a class of convolution-type schemes is $\alpha$-robust, i.e., the factor will not blowup while $\alpha\to 1^-$ with general nonuniform time steps even when rather general initial regularity condition is considered., Comment: 21 pages

A rotation-two-component Camassa-Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are $O(\tau^2 + h^2)$ for the velocity in the $L^\infty$-norm and the surface elevation in the $L^2$-norm, where $\tau$ denotes the temporal stepsize and $h$ the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Comparing with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values.

We introduce and analyze a robust nonconforming finite element method for a three dimensional singularly perturbed quad-curl model problem. For the solution of the model problem, we derive proper a priori bounds, based on which, we prove that the proposed finite element method is robust with respect to the singular perturbation parameter $\varepsilon$ and the numerical solution is uniformly convergent with order $h^{1/2}$. In addition, we investigate the effect of treating the second boundary condition weakly by Nitsche's method. We show that such a treatment leads to sharper error estimates than imposing the boundary condition strongly when the parameter $\varepsilon< h$. Finally, numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.

We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem. The proposed finite element spaces are subspaces of $\pmb{H}(\mathrm{curl})$, but not of $\pmb{H}(\mathrm{grad}~\mathrm{curl})$, which are different from the existing nonconforming ones. The well-posedness of the discrete problem is proved and optimal error estimates in discrete $\pmb{H}(\mathrm{grad}~\mathrm{curl})$ norm, $\pmb{H}(\mathrm{curl})$ norm and $\pmb{L}^2$ norm are derived. Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.

We propose two families of nonconforming elements on cubical meshes: one for the $-\text{curl}\Delta\text{curl}$ problem and the other for the Brinkman problem. The element for the $-\text{curl}\Delta\text{curl}$ problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter $\nu$. The lowest-order elements for the $-\text{curl}\Delta\text{curl}$ and the Brinkman problems have 48 and 30 degrees of freedom, respectively. The two families of elements are subspaces of $H(\text{curl};\Omega)$ and $H(\text{div};\Omega)$, and they, as nonconforming approximation to $H(\text{gradcurl};\Omega)$ and $[H^1(\Omega)]^3$, can form a discrete Stokes complex together with the Lagrange element and the $L^2$ element.

Using newly developed ${\bf H}(\mathrm{curl}^2)$ conforming elements, we solve the Maxwell's transmission eigenvalue problem. Both real and complex eigenvalues are considered. Based on the fixed-point weak formulation with reasonable assumptions, the optimal error estimates for numerical eigenvalues and eigenfunctions (in the ${\bf H}(\mathrm{curl}^2)$-norm and ${\bf H}(\mathrm{curl})$-semi-norm) are established.

An hp-version interior penalty discontinuous Galerkin (IPDG) method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive an error estimate in a mesh-dependent norm, which is optimal with respect to h but has different p-version error bounds under conforming and nonconforming tetrahedron meshes. The hp-version discrete compactness of the DG space is established for the convergence proof. The performance of the method is demonstrated by numerical experiments using conforming/nonconforming meshes and h-version/p-version refinement. The optimal h-version convergence rate and the exponential p-version convergence rate are observed.

In this work, we analyze the three-step backward differentiation formula (BDF3) method for solving the Allen-Cahn equation on variable grids. For BDF2 method, the discrete orthogonal convolution (DOC) kernels are positive, the stability and convergence analysis are well established in [Liao and Zhang, \newblock Math. Comp., \textbf{90} (2021) 1207--1226; Chen, Yu, and Zhang, \newblock SIAM J. Numer. Anal., Major Revised]. However, the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial, since the DOC kernels are not always positive. By developing a novel spectral norm inequality, the unconditional stability and convergence are rigorously proved under the updated step ratio restriction $r_k:=\tau_k/\tau_{k-1}\leq 1.405$ (compared with $r_k\leq 1.199$ in [Calvo and Grigorieff, \newblock BIT. \textbf{42} (2002) 689--701]) for BDF3 method. Finally, numerical experiments are performed to illustrate the theoretical results. To the best of our knowledge, this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.

We investigate numerical solutions of high order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite elements in complexes solve the problems correctly. To explain the numerical results, we clarify the cohomological structures in high order curl problems by relating the partial differential equations to the Hodge-Laplacian boundary problems of the gradcurl-complexes., Comment: 23 pages, 7 figures

Vibrations of structures subjected to concentrated point loads have many applications in mechanical engineering. Experiments are expensive and numerical methods are often used for simulations. In this paper, we consider the plate vibration with nonlinear dependence on the eigen-parameter. The problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function. The Bogner-Fox-Schmit element is used for the discretization and the spectral indicator method is employed to compute the eigenvalues. The convergence is proved using the abstract approximation theory of Karma. Numerical examples are presented for validations.

It is well known that the usual mixed method for solving the biharmonic eigenvalue problem by decomposing the operator into two Laplacians may generate spurious eigenvalues on non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. A priori error estimates for both eigenvalues and eigenfunctions on quasi-uniform meshes are obtained; in particular, a convergence rate of $\mathcal{O}({h}^{2\alpha})$ ($ 0<\alpha<\pi/\omega$, $\omega > \pi$ is the angle of the reentrant corner) is proved for the linear finite element. Surprisingly, numerical evidence demonstrates a $\mathcal{O}({h}^{2})$ convergent rate for the quasi-uniform mesh with the regular refinement strategy even on non-convex polygonal domains.

In this paper, we propose a sparse spectral-Galerkin approximation scheme for solving the second-order partial differential equations on an arbitrary tetrahedron. Generalized Koornwinder polynomials are introduced on the reference tetrahedron as basis functions with their various recurrence relations and differentiation properties being explored. The method leads to well-conditioned and sparse linear systems whose entries can either be calculated directly by the orthogonality of the generalized Koornwinder polynomials for differential equations with constant coefficients or be evaluated efficiently via our recurrence algorithm for problems with variable coefficients. Clenshaw algorithms for the evaluation of any polynomial in an expansion of the generalized Koornwinder basis are also designed to boost the efficiency of the method. Finally, numerical experiments are carried out to illustrate the effectiveness of the proposed Koornwinder spectral method., Comment: 29 pages, 24 figures

In this paper, we propose an $H(\text{curl}^2)$-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. $H(\text{curl}^2)$-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is astonishing that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our $H(\text{curl}^2)$-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

In this paper, a new $C^1$-conforming Petrov-Galerkin method for convection-diffusion equations is designed and analyzed. The trail space of the proposed method is a $C^1$-conforming ${\mathbb Q}_k$ (i.e., tensor product of polynomials of degree at most $k$) finite element space while the test space is taken as the $L^2$ (discontinuous) piecewise ${\mathbb Q}_{k-2}$ polynomial space. Existence and uniqueness of the numerical solution is proved and optimal error estimates in all $L^2, H^1, H^2$-norms are established. In addition, superconvergence properties of the new method are investigated and superconvergence points/lines are identified at mesh nodes (with order $2k-2$ for both function value and derivatives), at roots of a special Jacobi polynomial, and at the Lobatto lines and Gauss lines with rigorous theoretical analysis. In order to reduce the global regularity requirement, interior a priori error estimates in the $L^2, H^1, H^2$-norms are derived. Numerical experiments are presented to confirm theoretical findings.

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without time-step dependent on the spatial mesh size. The key is to estimate directly the solution bounds in the $H^2$-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence, and optimal error estimates, respectively, of the proposed fully discrete schemes., Comment: 21 pages, 4 figures

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn-Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise $L^2$ norm error estimate is established at the first time under an updated step-ratio restriction $0

In this paper, we study the biharmonic equation with the Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the 4th-order problem into a system of Poisson equations. Different from the usual mixed method that leads to two Poisson problems but only applies to convex domains, the proposed decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and non-convex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulted system. In addition, we derive the optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings., Comment: 19 pages, 5 figures

This article devotes to developing robust but simple correction techniques and efficient algorithms for a class of second-order time stepping methods, namely the shifted fractional trapezoidal rule (SFTR), for subdiffusion problems to resolve the initial singularity and nonlocality. The stability analysis and sharp error estimates in terms of the smoothness of the initial data and source term are presented. As a byproduct in numerical tests, we find amazingly that the Crank-Nicolson scheme ($\theta=\frac{1}{2}$) without initial corrections can restore the optimal convergence rate for the subdiffusion problem with smooth initial data and source terms. To deal with the nonlocality, fast algorithms are considered to reduce the computational cost from $O(N^2)$ to $O(N \log N)$ and save the memory storage from $O(N)$ to $O(\log N)$, where $N$ denotes the number of time levels. Numerical tests are performed to verify the sharpness of the theoretical results and confirm the efficiency and accuracy of initial corrections and the fast algorithms.

We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space. In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively. As a consequence, we obtain gradcurl-conforming finite elements and inf-sup stable Stokes pairs on tetrahedra which fit into complexes. We show that the new elements lead to convergent algorithms for solving a gradcurl model problem as well as solving the Stokes system with precise divergence-free condition. We demonstrate the validity of the algorithms by numerical experiments., Comment: 27 pages, 1 figure

We consider the calculation of the band structure of frequency dependent photonic crystals. The associated eigenvalue problem is nonlinear and it is challenging to develop effective convergent numerical methods. In this paper, the band structure problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Lagrange finite elements are used to discretize the operator function. Then the convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions. A spectral indicator method is developed to practically compute the eigenvalues. Numerical examples are presented to validate the theory and show the effectiveness of the proposed method.

In [24], we proposed H(curl^2)-conforming elements on both a triangle and a rectangle. This family of elements provides a brand new method to solve the quad-curl problem in 2 dimensions. In this paper, we turn our focus to 3 dimensions and construct an H(curl^2)-conforming tetrahedral finite element. The newly proposed element has been proved to have the optimal interpolation error estimate. Having tetrahedral elements, we can solve the quad-curl problem in any Lipschitz domain by conforming finite element method. We also provide several numerical examples of using our element to solve the quad-curl problem. The results of the numerical experiments show the effectiveness and correctness of our element., Comment: 6 figures;31 pages

Several smooth finite element de Rham complexes are constructed in three-dimensional space, which yield three families of grad-div conforming finite elements. The simplest element has only 8 degrees of freedom (DOFs) for a tetrahedron and 14 DOFs for a cuboid. These elements naturally lead to conforming approximations to quad-div problems. Numerical experiments for each family validate the correctness and efficiency of the elements for solving the quad-div problem., Comment: 28 pages; 6 figures. arXiv admin note: text overlap with arXiv:2004.12507

In this paper, we propose a new family of H(curl^2)-conforming elements for the quad-curl eigenvalue problem in 2D. The accuracy of this family is one order higher than that in [32]. We prove a priori and a posteriori error estimates. The a priori estimate of the eigenvalue with a convergence order 2(s-1) is obtained if the eigenvector u\in H^{s+1}(\Omega). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the eigenvector in an energy norm and an upper bound for the eigenvalues. Numerical examples are presented for validation., Comment: 26 pages, 9 figures

We construct smooth finite element de Rham complexes in two space dimensions. This leads to three families of curl-curl conforming finite elements, two of which contain two existing families. The simplest triangular and rectangular finite elements have only 6 and 8 degrees of freedom, respectively. Numerical experiments for each family demonstrate the convergence and efficiency of the elements for solving the quad-curl problem., Comment: 24 pages, 2 figures

In this paper, we present and study $C^1$ Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree $k$ ($\ge 3$) for one-dimensional elliptic equations. We prove that, the solution and its derivative approximations converge with rate $2k-2$ at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree $k+1$ in each element, the first-order derivative approximation is superconvergent at all interior $k-2$ Lobatto points, and the second-order derivative approximation is superconvergent at $k-1$ Gauss points, with an order of $k+2$, $k+1$, and $k$, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in $H^2$, $H^1$, and $L^2$ norms. All theoretical findings are confirmed by numerical experiments.

In this paper, we propose a new finite element approach, which is different than the classic Babuska-Osborn theory, to approximate Dirichlet eigenvalues. The Dirichlet eigenvalue problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Using conforming finite elements, the convergence is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed to compute the eigenvalues. A numerical example is presented to validate the theory.

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order temporal convergence can be recovered if almost all of time-step ratios $r_k\le 1+\sqrt{2}$ or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity at the discrete levels. An example is included to support our analysis., Comment: 20 pages

We apply two families of novel fractional $\theta$-methods, the FBT-$\theta$ and FBN-$\theta$ methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional $\theta$-methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with $O(\tau^2+h^{r+1})$ for smooth solutions, where $\tau$ is the time mesh size and $h$ is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time., Comment: 24 pages, 9 figures

The convolution quadrature theory is a systematic approach to analyse the approximation of the Riemann-Liouville fractional operator $I^{\alpha}$ at node $x_{n}$. In this paper, we develop the shifted convolution quadrature ($SCQ$) theory which generalizes the theory of convolution quadrature by introducing a shifted parameter $\theta$ to cover as many numerical schemes that approximate the operator $I^{\alpha}$ with an integer convergence rate as possible. The constraint on the parameter $\theta$ is discussed in detail and the phenomenon of superconvergence for some schemes is examined from a new perspective. For some technique purposes when analysing the stability or convergence estimates of a method applied to PDEs, we design some novel formulas with desired properties under the framework of the $SCQ$. Finally, we conduct some numerical tests with nonsmooth solutions to further confirm our theory., Comment: 21 pages, 15 figures

In this article, we introduce two families of novel fractional $\theta$-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator $\mathit{I}^{\alpha}$ with a second order convergence rate. A new fractional BT-$\theta$ method connects the fractional BDF2 (when $\theta=0$) with fractional trapezoidal rule (when $\theta=1/2$), and another novel fractional BN-$\theta$ method joins the fractional BDF2 (when $\theta=0$) with the second order fractional Newton-Gregory formula (when $\theta=1/2$). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different $\theta$-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional $\theta$-methods are A($\vartheta$)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis., Comment: 16 pages, 10 figures

Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional $H^1$-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accuracy). To recover the theoretical accuracy in time, we propose an improved discrete Gr\"{o}nwall inequality and apply it to the well-known L1 formula and a fractional Crank-Nicolson scheme. With the help of a time-space error-splitting technique and the global consistency analysis, sharp $H^1$-norm error estimates of the two nonuniform approaches are established for a reaction-subdiffusion problems. Numerical experiments are included to confirm the sharpness of our analysis., Comment: 22 pages, 8 tables

In this paper, we first construct the $H^2$(curl)-conforming finite elements both on a rectangle and a triangle. They possess some fascinating properties which have been proven by a rigorous theoretical analysis. Then we apply the elements to construct a finite element space for discretizing quad-curl problems. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^{k-1})$ in the $H^2$(curl) norm are established. Numerical experiments are provided to confirm our theoretical results.

In this paper, we develop a straightforward $C^0$ linear finite element method for sixth-order elliptic equations. The basic idea is to use gradient recovery techniques to generate higher-order numerical derivatives from a $C^0$ linear finite element function. Both theoretical analysis and numerical experiments show that the proposed method has the optimal convergence rate under the energy norm. The method avoids complicated construction of conforming $C^2$ finite element basis or nonconforming penalty terms and has a low computational cost.

In this paper, we study convergence and superconvergence theory of integer and fractional derivatives of the one-point and the two-point Hermite interpolations. When considering the integer-order derivative, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are $O(N^{-2})$ and $O(N^{-1.5})$, respectively, better than the global rate for the one-point and two-point interpolations. Here $N$ represents the degree of interpolation polynomial. It is proved that the $\alpha$-th fractional derivative of $(u-u_N)$ with $k<\alpha

In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $\alpha>-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator, and a Sturm-Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalisation of orthogonal {\em ball polynomials} in primitive variables with a tuning parameter $c>0$, through a "perturbation" of the Sturm-Liouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutions., Comment: 25 pages

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.

Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin method. Some numerical examples are provided to validate our theoretical analysis., Comment: 4 figure2, 20 pages

We propose and analyze an efficient spectral-Galerkin approximation for the Maxwell transmission eigenvalue problem in spherical geometry. Using a vector spherical harmonic expansion, we reduce the problem to a sequence of equivalent one-dimensional TE and TM modes that can be solved individually in parallel. For the TE mode, we derive associated generalized eigenvalue problems and corresponding pole conditions. Then we introduce weighted Sobolev spaces based on the pole condition and prove error estimates for the generalized eigenvalue problem. The TM mode is a coupled system with four unknown functions, which is challenging for numerical calculation. To handle it, we design an effective algorithm using Legendre-type vector basis functions. Finally, we provide some numerical experiments to validate our theoretical results and demonstrate the efficiency of the algorithms., Comment: 22 pages, 8 figures

We study superconvergence property of the linear finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The $H^1$-error estimate with explicit dependence on the wave number $k$ {is} derived. First, we prove that under the assumption $k(kh)^2\leq C_0$ ($h$ is the mesh size) and certain mesh condition, the estimate between the finite element solution and the linear interpolation of the exact solution is superconvergent under the $H^1$-seminorm, although the pollution error still exists. Second, we prove a similar result for the recovered gradient by PPR and found that the PPR can only improve the interpolation error and has no effect on the pollution error. Furthermore, we estimate the error between the finite element gradient and recovered gradient and discovered that the pollution error is canceled between these two quantities. Finally, we apply the Richardson extrapolation to recovered gradient and demonstrate numerically that PPR combined with the Richardson extrapolation can reduce the interpolation and pollution errors simultaneously, and therefore, leads to an asymptotically exact {\it a posteriori} error estimator. All theoretical findings are verified by numerical tests., Comment: 25 pages, 16 figures. arXiv admin note: substantial text overlap with arXiv:1612.03386

The contribution of this paper contains two parts: first, we prove a supercloseness result for the partially penalized immersed finite element (PPIFE) method in [T. Lin, Y. Lin, and X. Zhang, SIAM J. Numer. Anal., 53 (2015), 1121--1144]; then based on the supercloseness result, we show that the gradient recovery method proposed in our previous work [H. Guo and X. Yang, arXiv: 1608.00063] can be applied to the PPIFE method and the recovered gradient converges to the exact gradient with a superconvergent rate $\mathcal{O}(h^{3/2})$. Hence, the gradient recovery method provides an asymptotically exact a posteriori error estimator for the PPIFE method. Several numerical examples are presented to verify our theoretical result., Comment: 19 pages, 3 figures

In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in H1 and L2 norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order p+2 at the roots of generalized Lobatto polynomials, and the flux is superconvergent of order p+1 at generalized Gauss points on each element including the interface element. Furthermore, for diffusion interface problems, the convergence rates for IFVM solution at the mesh points and the flux at generalized Gaussian points can both be raised to 2p. These superconvergence results are consistent with those for the standard finite volume methods. Numerical examples are provided to confirm our theoretical analysis.

We study superconvergence property of the linear discontinuous Galerkin finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The error estimate with explicit dependence on the wave number $k$, the penalty parameter $\mu$ and the mesh condition parameter $\alpha$ is derived. First, we prove that under the assumption $k(kh)^2\leq C_0$ ($h$ is the mesh size) and certain mesh condition, the estimate between the finite element solution and the linear interpolation of the exact solution is superconvergent under the $\norme{\cdot}$-seminorm. Second, we prove a superconvergence result for the recovered gradient by PPR. Furthermore, we estimate the error between the finite element gradient and recovered gradient, which motivate us to define the a posteriori error estimator. Finally, Some numerical examples are provided to confirm the theoretical results of superconvergence analysis. All theoretical findings are verified by numerical tests., Comment: 22 pages, 10 fiures