Optimal Order Of Convergence Research Articles

An unfitted HDG method for a distributed optimal control problem

We analyze a high order hybridizable discontinuous Galerkin (HDG) method for an optimal control problem where the computational mesh does not necessarily fit the domain. The method is based on transferring the boundary data to the computational boundary by integrating the approximation of the gradient. We prove optimal order of convergence in the L2-norm for all the variables of the state and adjoint problems, and the control variable as well. More precisely, order hk+1 if the local discrete spaces are constructed using polynomials of degree at most k on a triangulation of meshsize h. We present numerical experiments illustrating the performance of method.

Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2-1σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L2-norm and H1-norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H=h12 and H=hr2r+2, respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods.

An Adaptive and Quasi-periodic HDG Method for Maxwell’s Equations in Heterogeneous Media

With the aim to continue developing a hybridizable discontinuous Galerkin (HDG) method for problems arisen from photovoltaic cells modeling, in this manuscript we consider the time harmonic Maxwell’s equations in an inhomogeneous bounded bi-periodic domain with quasi-periodic conditions on part of the boundary. We propose an HDG scheme where quasi-periodic boundary conditions are imposed on the numerical trace space. Under regularity assumptions and a proper choice of the stabilization parameter, we prove that the approximations of the electric and magnetic fields converge, in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2$$\\end{document}-norm, to the exact solution with order hk+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1}$$\\end{document} and hk+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1/2}$$\\end{document}, resp., where h is the meshsize and k the polynomial degree of the discrete spaces. Although, numerical evidence suggests optimal order of convergence for both variables. An a posteriori error estimator for an energy norm is also proposed. We show that it is reliable and locally efficient under certain conditions. Numerical examples are provided to illustrate the performance of the quasi-periodic HDG method and the adaptive scheme based on the proposed error indicator.

Sparse precision matrix estimation under lower polynomial moment assumption

Precision matrix (inverse covariance matrix) estimation is a rising challenge in contemporary applications while dealing with high‐dimensional data. This paper focuses on large‐scale precision matrix of the random vector that only has lower polynomial moments. We mainly investigate upper bounds of the proposed estimator under the spectral norm in terms of the probability and mean estimation respectively. It is shown that the data‐driven estimator is fully adaptive and achieves the same optimal convergence order as under Gaussian assumption on the data. Simulation studies further support our theoretical claims.

Convergent finite element methods for the perfect conductivity problem with close-to-touching inclusions

Abstract In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general geometry still remains open in three dimensions. We address this problem by establishing new asymptotic estimates for the second-order partial derivatives of the solution with explicit dependence on the distance $\varepsilon $ between the inclusions, and use the asymptotic estimates to design a class of graded meshes and finite element spaces to solve the perfect conductivity problem with possibly close-to-touching inclusions. In particular, we propose a special finite element basis function that resolves the asymptotic singularity of the solution by making the interpolation error bounded in $W^{1,\infty }$ in a neighborhood of the close-to-touching point, even though the solution itself is blowing up in $W^{1,\infty }$. This is crucial in the error analysis for the numerical approximations. We prove that the proposed method yields optimal-order convergence in the $H^1$ norm, uniformly with respect to the distance $\varepsilon $ between the inclusions, in both two and three dimensions for general convex smooth inclusions, which are possibly close-to-touching. Numerical experiments are presented to support the theoretical analysis and to illustrate the convergence of the proposed method for different shapes of inclusions in both two- and three-dimensional domains.

A novel high resolution fifth-order weighted essentially non-oscillatory scheme for solving hyperbolic equations

Weighted essentially non-oscillator (WENO) scheme is popular in solving hyperbolic conservation equations. In this work, a novel high resolution fifth-order WENO scheme is presented to improve the performance for capturing of discontinuities and complex structures. A local smoothness indicator ISk with compact form is adopted to the WENO scheme of Borges (WENO-Z). Then, a new global smooth indicator η containing information of all sub-templates is constructed to improve the resolution. Furthermore, the nonlinear weight strategy is optimized, and a new term coefficient λ is proposed, which can adjust the weight adaptively with the change of flow field. The accuracy tests verify that the proposed scheme can achieve the optimal order convergence at the critical points of orders 0, 1, and 2, and the weight analysis indicates that the proposed scheme increases the weights of less smooth sub-templates and gives full play to the potential advantages of WENO-Z. Finally, several typical numerical tests show the proposed scheme can significantly improve the resolution and robustness of WENO scheme. This proposed WENO scheme can capture the discontinuities more sharply and can clearly identify vortex structures and fine vesicles compared with other three WENO schemes.

Implicit Runge‐Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian

AbstractAn efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.

Three Low Order H-Curl-Curl Finite Elements on Triangular Meshes

Abstract We construct three H-curl-curl finite elements. The P 2 P_{2} and P 3 P_{3} vector finite element spaces are both enriched by one common P 4 P_{4} bubble and their local degrees of freedom are 13 and 21, respectively. As there does not exist any P 1 P_{1} H-curl-curl conforming finite element, the P 1 P_{1} H-curl-curl nonconforming finite element is constructed with three additional P 4 P_{4} bubbles. Numerical tests are presented, confirming the conformity and the optimal order of convergence.

A hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations

A new fourth order highly efficient Exponential Time Differencing Runge-Kutta type hybrid time stepping method is developed by hybriding two fourth order methods. The new hybrid method takes advantages of a positivity preserving L-stable method to damp unnatural oscillations due to low regularity of the initial data and of a computationally efficient A-stable method to achieve optimal order convergence. Computational efficiency and stability of the new method is further enhanced by applying a splitting technique which makes it possible to implement the method on parallel processors. A hybriding criteria is presented and an algorithm based on the hybrid method is developed. The method is implemented to solve two two-dimensional problems having Riesz space distributed order diffusion and nonlinear reaction terms, an Allen-Cahn equation with a cubic nonlinearity and an Enzyme Kinetics equation with a rational nonlinearity. Fourth-order temporal convergence is obtained through numerical experiments. A third test problem with exact solution available is also considered and both spatial as well as temporal orders of convergence are computed. High computational efficiency of the method is shown by recording the CPU time. Numerical solutions using different order strength distribution functions are also presented.

A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn–Hilliard–Navier–Stokes system

We propose and analyze a structure-preserving space-time variational discretization method for the Cahn–Hilliard–Navier–Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration-dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.

The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality

We develop an interior penalty virtual element method (IPVEM) for solving a Kirchhoff plate contact problem, which can be described by a fourth-order elliptic hemivariational inequality (HVI). The virtual element space in IPVEM is constructed by modifying the H2-conforming virtual element space, and the number of degrees of freedom is greatly reduced. To force the C1 continuity, the interior penalty scheme is adopted, that is, we introduce a penalty term for the jump of normal derivatives on mesh edges and two additional terms to guarantee the consistency and symmetry of the scheme. With certain assumptions, the well-posedness of the discrete problem is proved. Furthermore, a priori error estimation is established for the IPVEM for the fourth-order elliptic HVI, and we show that the lowest-order VEM achieves optimal convergence order. Finally, some numerical examples are presented to support the theoretical results.

A surface finite element method for the Navier–Stokes equations on evolving surfaces

AbstractWe introduce a surface finite element method for the numerical solution of Navier–Stokes equations on evolving surfaces with a prescribed deformation of the surface in the normal direction. The method is based on approaches for the full surface Navier–Stokes equations in the context of fluid‐deformable surfaces and adds a penalization of the normal component of the velocity. Numerical results demonstrate the same optimal order of convergence as proposed for surface (Navier–)Stokes equations on stationary surfaces. The approach is applied to high‐resolution three‐dimensional scans of clothed bodies in motion to provide interactive virtual fluid‐like clothing.

Improved enrichments and numerical integrations in SGFEM for interface problems

Generalized/extended finite element methods (GFEM/XFEM) for an interface problem are generally enriched by a distance function to the interface to handle discontinuities across the interface. Evaluations of the distance function and its gradients are needed for assembling stiffness matrices, which requires certain computational geometry algorithms. The computational complexity grows if the number of integration points is increased. Besides, the mathematical analysis on the effect of numerical integration has not been established in the field of GFEM/XFEM yet. This study designs an improved enriched function for a stable GFEM (SGFEM, a stable version of GFEM) based on the distance function, which only requires evaluating the distance function at nodes of elements that interact the interface (the gradients of distance function are not needed). The optimal convergence order (in the energy norm) of the SGFEM with such an improved enriched function is proven. In turn, we propose a perturbed variational formula, which can be integrated exactly by simple Gaussian rules designed in this paper. We prove that the SGFEM based on the improved enriched function and the perturbed variational formula (exactly integrated by the numerical integration) achieves the optimal convergence order for the interface problem. Namely, we develop a numerical integration rule for the SGFEM such that the optimal convergence order of SGFEM is maintained for the interface problem. The theoretical achievements are verified by numerical experiments.

Isogeometric analysis and Augmented Lagrangian Galerkin Least Squares Methods for residual minimization in dual norm

We explore how recent advances in Isogeometric analysis, Galerkin Least-Squares methods, and Augmented Lagrangian techniques can be applied to solve nonstandard problems, for which there is no classical stability theory, such as that provided by the Lax–Milgram lemma or the Banach-Necas-Babuska theorem. In particular, we consider continuation problems where a second-order partial differential equation with incomplete boundary data is solved given measurements of the solution on a subdomain of the computational domain. The use of higher regularity spline spaces leads to simplified formulations and potentially minimal multiplier space. We show that our formulation is inf-sup stable, and given appropriate a priori assumptions, we establish optimal order convergence.

On Two Competing Methods with Optimal Eighth Order Convergence

On Two Competing Methods with Optimal Eighth Order Convergence

On the reconstruction of functions from values at subsampled quadrature points

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with quasi-Monte Carlo (QMC) points with inherent structure and stable L 2 L_2 reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a reproducing kernel Hilbert space of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the d d -dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.

Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems

In this paper, we propose a weak Galerkin finite element method (WG-FEM) for solving two-point boundary value problems of convection-dominated type on a Bakhvalov-type mesh. A special interpolation operator which has a simple representation and can be easily extended to higher dimensions is introduced for convection-dominated problems. A robust optimal order of uniform convergence has been proved in the energy norm with this special interpolation using piecewise polynomials of degree $k\geq 1$ on interior of the elements and piecewise constant on the boundary of each element. The proposed finite element scheme is {parameter-free formulation} and since the interior degrees of freedom can be eliminated efficiently from the resulting discrete system, the number of unknowns of the proposed method is comparable with the standard finite element methods. An optimal order of uniform convergence is derived on Bakhvalov-type mesh. Finally, numerical experiments are given to support the theoretical findings and to show the efficiency of the proposed method.

Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient

In this paper, a fully discrete finite element scheme is established with L1 approximation for solving the nonlinear time-fractional wave equation with a variable coefficient. The stability of the scheme is analyzed and the optimal error estimates are derived. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete finite element scheme. With the proposed technique, a small-scale nonlinear problem is calculated by iteration in a coarse-grid space with mesh size H, and then a linearized problem is solved in a fine-grid space with mesh size h, H≫h. This technique not only maintains the numerical precision, but also saves a lot of computing time. The stability and optimal convergence order are considered. Using the Taylor expansion, the second two-grid algorithm is constructed to improve the optimal convergence order. The similar properties are discussed in detail. Numerical experiments are provided to illustrate the theoretical analysis and test the performance of the proposed algorithms.

A Stable and Convergent Hybridized Discontinuous Galerkin Method for Time-Fractional Telegraph Equations

In this paper, we extend the application of the hybridized discontinuous Galerkin (HDG) method to solve time-fractional telegraph equations. In fact, we use an HDG method for space discretization and L1 and L2 finite difference schemes using non-uniform meshes for time discretization. Thanks to a special kind of discrete Gronwall inequality, we prove that the HDG method is unconditionally stable and it is convergent with the optimal spatial order of convergence. Two numerical experiments are tested to confirm the theoretical results.

Optimal eighth-order multiple root finding iterative methods using bivariate weight function

In this contribution, a novel eighth-order scheme is presented for solving nonlinear equations with multiple roots. The proposed scheme comprises of three steps with the modified Newton method as its first step followed by two weighted Newton steps involving one univariate and one bivariate function respectively. Analysis of convergence confirms that the presented scheme obtains optimal computational order of convergence. The efficiency of presented scheme is compared numerically with recent eighth-order methods. Functions like population growth problem, Newton’s beam problem, etc., have been considered for numerical experimentation. For the comparative study in the complex plane, we employed the concept of basins of attraction.