# American Institute of Mathematical Sciences

eISSN:
2688-1594

All Issues

## Electronic Research Archive

### Special issue on higher-order numerical methods for PDEs & applications

Yanping Chen, School of Mathematical Sciences, South China Normal University, No.55 Zhongshan Avenue West, Tianhe District, Guangzhou 510631, China yanpingchen@scnu.edu.cn

Kai Jiang, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan province, 411105, China kaijiang@xtu.edu.cn

Joaquim Rigola, Heat and Mass Transfer Technological Center (CTTC)Universitat Politècnica de Catalunya - Barcelona Tech (UPC)ESEIAAT, C/ Colom, 11 - 08222 Terrassa (Barcelona), Spain quim@cttc.upc.edu

2020, 28(1): 405-421 doi: 10.3934/era.2020023 +[Abstract](11)+[HTML](9) +[PDF](1697.67KB)
Abstract:

Granular materials are heterogenous grains in contact, which are ubiquitous in many scientific and engineering applications such as chemical engineering, fluid mechanics, geomechanics, pharmaceutics, and so on. Granular materials pose a great challenge to predictability, due to the presence of critical phenomena and large fluctuation of local forces. In this paper, we consider the quasi-static simulation of the dense granular media, and investigate the performances of typical minimization algorithms such as conjugate gradient methods and quasi-Newton methods. Furthermore, we develop preconditioning techniques to enhance the performance. Those methods are validated with numerical experiments for typical physically interested scenarios such as the jamming transition, the scaling law behavior close to the jamming state, and shear deformation of over jammed states.

2020, 28(2): 821-836 doi: 10.3934/era.2020042 +[Abstract](11)+[HTML](5) +[PDF](377.34KB)
Abstract:

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

2020, 28(2): 897-910 doi: 10.3934/era.2020047 +[Abstract](19)+[HTML](8) +[PDF](310.06KB)
Abstract:

In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the \begin{document}$H^1$\end{document}-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

2020, 28(2): 911-933 doi: 10.3934/era.2020048 +[Abstract](17)+[HTML](11) +[PDF](1531.97KB)
Abstract:

This paper is concerned with a \begin{document}$C^0P_2$\end{document} time-stepping virtual element method (VEM) for solving linear wave equations on polygonal meshes. The spatial discretization is carried out by the VEM while the temporal discretization is obtained based on the \begin{document}$C^0P_2$\end{document} time-stepping approach, leading to a fully discrete method. The error estimates in the \begin{document}$H^1$\end{document} semi-norm and \begin{document}$L^2$\end{document} norm are derived after detailed derivation. Finally, the numerical performance and efficiency of the proposed method is illustrated by several numerical experiments.

2020, 28(2): 935-949 doi: 10.3934/era.2020049 +[Abstract](17)+[HTML](10) +[PDF](434.86KB)
Abstract:

A multiscale finite element method is proposed for addressing a singularly perturbed convection-diffusion model. In reference to the a priori estimate of the boundary layer location, we provide a graded recursion for the mesh adaption. On this mesh, the multiscale basis functions are able to capture the microscopic boundary layers effectively. Then with a global reduction, the multiscale functional space may efficiently reflect the macroscopic essence. The error estimate for the adaptive multiscale strategy is presented, and a high-order convergence is proved. Numerical results validate the robustness of this novel multiscale simulation for singular perturbation with small parameters, as a consequence, high accuracy and uniform superconvergence are obtained through computational reductions.

Kai Jiang and
2020, 28(2): 1077-1093 doi: 10.3934/era.2020059 +[Abstract](27)+[HTML](19) +[PDF](3164.56KB)
Abstract:

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the \begin{document}$L^2$\end{document} gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.

2020, 28(3): 1161-1189 doi: 10.3934/era.2020064 +[Abstract](23)+[HTML](12) +[PDF](436.96KB)
Abstract:

In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the \begin{document}$L^{\infty}$\end{document}- and \begin{document}$L^{2}_{\omega^{\alpha, \beta}}$\end{document}-norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.

2020, 28(3): 1191-1205 doi: 10.3934/era.2020065 +[Abstract](9)+[HTML](8) +[PDF](361.28KB)
Abstract:

In this paper, we construct a modular grad-div stabilization method for the Navier-Stokes/Darcy model, which is based on the first order Backward Euler scheme. This method does not enlarge the accuracy of numerical solution, but also can improve mass conservation and relax the influence of parameters. Herein, we give stability analysis and error estimations. Finally, by some numerical experiment, the scheme our proposed is shown to be valid.

2020, 28(3): 1207-1225 doi: 10.3934/era.2020066 +[Abstract](17)+[HTML](7) +[PDF](1610.81KB)
Abstract:

In this work, a multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is proposed to solve a coupled chemotaxis-fluid model. In the evolution equation of the proposed LBM, Beam-Warming (B-W) scheme is used to enhance the numerical stability. In numerical experiments, at first, the comparison between the classical LBM and the present LBM with B-W scheme is carried out by simulating blow up phenomenon of the Keller-Segel (K-S) model. Numerical results show that the stability of the present LBM with B-W scheme is better than the classical one. Then, the second order convergence rate of the proposed B-W scheme is verified in the numerical study of the coupled Navier-Stokes (N-S) K-S model. Finally, through solving the coupled chemotaxis-fluid model, the formation of falling bacterial plumes is numerically investigated. Numerical results agree well with existing ones in the literature.

2020, 28(4): 1439-1457 doi: 10.3934/era.2020076 +[Abstract](15)+[HTML](16) +[PDF](466.79KB)
Abstract:

In this paper, an adaptive grid method is proposed to solve one-dimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arc-length function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.

2020, 28(4): 1459-1486 doi: 10.3934/era.2020077 +[Abstract](16)+[HTML](12) +[PDF](2201.39KB)
Abstract:

This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by \begin{document}$L^2$\end{document}-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.

2020, 28(4): 1487-1501 doi: 10.3934/era.2020078 +[Abstract](16)+[HTML](11) +[PDF](5508.22KB)
Abstract:

We numerically investigate the superconvergence property of the discontinuous Galerkin method by patch reconstruction. The convergence rate \begin{document}$2m+1$\end{document} can be observed at the grid points and barycenters in one dimensional case with uniform partitions. The convergence rate \begin{document}$m + 2$\end{document} is achieved at the center of the element faces in two and three dimensions. The meshes are uniformly partitioned into triangles/tetrahedrons or squares/hexahedrons. We also demonstrate the details of the implementation of the proposed method. The numerical results for all three dimensional cases are presented to illustrate the propositions.

2020, 28(4): 1503-1528 doi: 10.3934/era.2020079 +[Abstract](15)+[HTML](11) +[PDF](1371.93KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

Title change has delayed IF 2018  Impact Factor: 0.263