#### Table of Content

15 September 2022, Volume 40 Issue 5
TWO-GRID ALGORITHM OF H1-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS
Tianliang Hou, Chunmei Liu, Chunlei Dai, Luoping Chen, Yin Yang
2022, 40(5):  667-685.  DOI: 10.4208/jcm.2101-m2019-0159
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In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H1-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice h = H 2.
A DISCRETIZING LEVENBERG-MARQUARDT SCHEME FOR SOLVING NONLIEAR ILL-POSED INTEGRAL EQUATIONS
Rong Zhang, Hongqi Yang
2022, 40(5):  686-710.  DOI: 10.4208/jcm.2101-m2020-0218
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To reduce the computational cost, we propose a regularizing modified LevenbergMarquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.
HEAVY BALL FLEXIBLE GMRES METHOD FOR NONSYMMETRIC LINEAR SYSTEMS
Mei Yang, Ren-Cang Li
2022, 40(5):  711-727.  DOI: 10.4208/jcm.2101-m2019-0243
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Flexible GMRES (FGMRES) is a variant of preconditioned GMRES, which changes preconditioners at every Arnoldi step. GMRES often has to be restarted in order to save storage and reduce orthogonalization cost in the Arnoldi process. Like restarted GMRES, FGMRES may also have to be restarted for the same reason. A major disadvantage of restarting is the loss of convergence speed. In this paper, we present a heavy ball flexible GMRES method, aiming to recoup some of the loss in convergence speed in the restarted flexible GMRES while keep the benefit of limiting memory usage and controlling orthogonalization cost. Numerical tests often demonstrate superior performance of the proposed heavy ball FGMRES to the restarted FGMRES.
PENALTY-FACTOR-FREE STABILIZED NONCONFORMING FINITE ELEMENTS FOR SOLVING STATIONARY NAVIER-STOKES EQUATIONS
Linshuang He, Minfu Feng, Qiang Ma
2022, 40(5):  728-755.  DOI: 10.4208/jcm.2101-m2020-0156
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Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous P1 vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and L2-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of L2-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.
PRIMAL-DUAL PATH-FOLLOWING METHODS AND THE TRUST-REGION UPDATING STRATEGY FOR LINEAR PROGRAMMING WITH NOISY DATA
Xinlong Luo, Yiyan Yao
2022, 40(5):  756-776.  DOI: 10.4208/jcm.2101-m2020-0173
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In this article, we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem. For the rank-deficient problem with the small noisy data, we also give the preprocessing method based on the QR decomposition with column pivoting. Then, we prove the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, for some rankdeficient problems with or without the small noisy data from the NETLIB collection, we compare it with other two popular interior-point methods, i.e. the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment. Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.
ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA
Kai Wang, Na Wang
2022, 40(5):  777-793.  DOI: 10.4208/jcm.2101-m2020-0075
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This article concerns numerical approximation of a parabolic interface problem with general L2 initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k = 1, … , 6. To maintain high-order convergence in time for possibly nonsmooth L2 initial value, we modify the standard backward difference formula at the first k?1 time levels by using a method recently developed for fractional evolution equations. An error bound of O(tn?kτ k +tn?1h2| log h|) is established for the fully discrete finite element method for general L2 initial data.
A CHARACTERISTIC MIXED FINITE ELEMENT TWO-GRID METHOD FOR COMPRESSIBLE MISCIBLE DISPLACEMENT PROBLEM
Hanzhang Hu, Yanping Chen
2022, 40(5):  794-813.  DOI: 10.4208/jcm.2101-m2020-0277
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A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium. The concentration equation is treated by a mixed finite element method with characteristics (CMFEM) and the pressure equation is treated by a parabolic mixed finite element method (PMFEM). Two-grid algorithm is considered to linearize nonlinear coupled system of two parabolic partial differential equations. Moreover, the Lq error estimates are conducted for the pressure, Darcy velocity and concentration variables in the two-grid solutions. Both theoretical analysis and numerical experiments are presented to show that the two-grid algorithm is very effective.
ANALYSIS OF A MULTI-TERM VARIABLE-ORDER TIME-FRACTIONAL DIFFUSION EQUATION AND ITS GALERKIN FINITE ELEMENT APPROXIMATION
Huan Liu, Xiangcheng Zheng, Hongfei Fu
2022, 40(5):  814-834.  DOI: 10.4208/jcm.2102-m2020-0211
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In this paper, we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation, and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution. We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t = 0. More precisely, we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C2([0,T ]) in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness, otherwise the solution exhibits the same singular behavior like its constant-order counterpart. Based on these regularity results, we prove optimalorder convergence rate of the Galerkin finite element scheme. Furthermore, we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives. Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.