In this paper, we propose and analyze a modified adaptive two-grid (MATG) finite element method (FEM) for nonsymmetric or indefinite elliptic problems. In this method, we need to solve an extra symmetric positive definite (SPD) residual equation and correct the solution in the previous step before solving the SPD approximate problem. We construct a new residual-based a posteriori error estimator and prove its reliability. We also prove the contraction of the quasi-error and the convergence of the modified algorithm. At last, we report some numerical results to show the effectiveness and robustness of the proposed method.
This paper is to prove one-order superconvergence of both stress and displacement of a conforming symmetric mixed finite element on uniform n-square grids, for the linear elasticity equation in the Hellinger-Reissner variational formulation. Numerical examples on 2D and 3D uniform square grids are computed, verifying the theory.
In this work, modified Patankar-Runge-Kutta (MPRK) schemes up to order four are considered and equipped with a dense output formula of appropriate accuracy. Since these time integrators are conservative and positivity-preserving for any time step size, we impose the same requirements on the corresponding dense output formula. In particular, we discover that there is an explicit first-order formula. However, to develop a boot-strapping technique, we propose to use implicit formulae which naturally fit into the framework of MPRK schemes. In particular, if lower-order MPRK schemes are used to construct methods of higher order, the same can be done with the dense output formulae we propose in this work. We explicitly construct formulae up to order three and demonstrate how to generalize this approach as long as the underlying Runge-Kutta method possesses a dense output formula of appropriate accuracy. We also note that even though linear systems have to be solved to compute an approximation for intermediate points in time using these higher-order dense output formulae, the overall computational effort to reach a given number of approximations is reduced compared to using the scheme with a smaller step size. We support this fact and our theoretical findings by means of numerical experiments.
Based on the Toeplitz structure contained in the banded preconditioner with shift compensation (BSC preconditioner), the asymptotic singular value distribution of the BSC-preconditioned matrix for solving the space fractional diffusion equations with nonequal diffusion coefficients is analyzed by exploiting the theory of the generalized locally Toeplitz (GLT) sequence. The theoretical analysis illustrates that the conditioning of the BSC-preconditioned matrix is bounded by when is sufficiently close to 2, and is bounded by when is sufficiently close to 1. Numerical computation also verifies that the BSC preconditioner is robust for sufficiently approaching 1 or 2.
This work demonstrates a numerical scheme comprising the Crank-Nicolson difference scheme in the temporal direction and cubic-trigonometric splines in the spatial direction for solving the time-fractional damped convection-diffusion wave delay differential equation. This equation involves the reaction and damping term and the delay parameter applied in time in the reaction term. This type of delay differential equation was not explored earlier, so we present a numerical scheme that is second-order convergent in the spatial direction and order of accuracy in the temporal direction. The proposed method is proved to be unconditionally stable and convergent through rigorous analysis. Two test examples are solved to validate our theoretical findings and manifest the proficiency of the present numerical scheme.
In this article, for a degenerate inhomogeneous equation of even order with a fractional derivative in the sense of Caputo, a boundary value problem of the Robin problem type is studied. The solution is constructed in the form of a series of eigenfunctions of a one-dimensional spectral problem for a degenerate equation of even order. The existence and positivity of eigenvalues are shown by reducing the spectral problem to an equivalent integral equation with a symmetric kernel. Also, when constructing the posed problem, a boundary value problem for a one-dimensional fractional order equation was studied. Depending on the sign of the constant coefficient of the equation q, the necessary estimates for the solution were obtained. Sufficient conditions for the convergence of the series, which is a solution to the Robin problem, and the series obtained by differentiation are found. The uniqueness of the solution is demonstrated by the spectral method.
We develop a set of numerical approximations to discretize the non-isothermal incompressible hydrodynamic model for Rayleigh-Bénard convection (RBC), which ensures the negative energy dissipation rate with respect to adiabatic boundary conditions. Using the Crank-Nicolson (CN) method, the second-order backward difference method combined with the pressure-correction method, we propose two kinds of decoupled, linear, both second-order in time energy-dissipation-rate-preserving semi-discrete projection numerical algorithms. Meanwhile, the second-order fully discrete numerical algorithms are obtained by the use of a finite difference method on staggered grids in space. These numerical approximations are proved to preserve the property of the energy dissipation rate at the fully discrete level. Moreover, the numerical approximations are also unconditionally energy stable. The fast Fourier algorithm is applied to numerical implementation to enhance experimental efficiency. Convergence rate tests are conducted to verify the accuracy of the algorithms. Our simulations showcase that both hydrodynamic and thermal effects in resolving RBC within the non-isothermal hydrodynamic model of incompressible viscous fluid flow. In general, our structure-preserving projection schemes and implementation methods accurately and efficiently simulate RBC in nature.
We present a discrete exterior calculus (DEC) based on the discretization scheme for axisymmetric incompressible two-phase flows, in which the previous work [39] is extended to its axisymmetric version. We first transform the axisymmetric two-phase incompressible Navier-Stokes (NS) equations and the auxiliary conservative phase field (PF) equation into the exterior calculus framework using differential forms and exterior operators. Discretization of these exterior calculus equations is obtained using discrete differential forms and exterior operators. The PF variable, used to capture the interface between the two phases, varies from zero to unity, and preserving these bounds is desirable. Several verification and validation tests are presented to numerically confirm mass conservation, solution boundedness, and convergence properties. Various axisymmetric two-phase flow simulations, including a drop oscillation, a bubble bursting, a rising bubble, and a drop merger, with large density and viscosity ratios and surface tension, demonstrate the excellent performance of DEC dealing with axisymmetric two-phase flows.
In this paper, we consider deriving some new gradient-based iterative (GI)-like algorithms for solving a class of Sylvester tensor equations, which often arise from control systems and image processing. We first study the optimal parameter and the iteration matrix’s minimal spectral radius of the relaxed GI (RGI) algorithm proposed by Zhang and Wang (Taiwan J Math 26: 501–519, 2022) in terms of matricization of a tensor and straightening operator. Then based on the Jacobi method, by using the diagonal matrices to replace the system matrices in mode products contained in the RGI and the modified RGI (MRGI) algorithms (Taiwan J Math 26: 501–519, 2022), we design the Jacobi RGI (JRGI) and improved MRGI (IMRGI) algorithms for the Sylvester tensor equations, which require less computational load and are more efficient than the RGI and the MRGI ones, respectively. We deduce the sufficient convergence condition and quasi-optimal parameter of the JRGI algorithm, and sufficient and necessary conditions for the convergence of the IMRGI algorithm. Furthermore, we apply a new update strategy to the RGI algorithm and develop an updated RGI (URGI) algorithm for the Sylvester tensor equations, which is different from the MGI (Math Probl Eng 819479: 1–7, 2013) and the MRGI ones. The URGI algorithm takes full advantage of the latest computed results and returns better convergence behavior than RGI, MGI, and MRGI ones. Also, we prove that the proposed URGI algorithm is convergent under proper conditions. Finally, numerical experiments are performed to verify that the proposed algorithms are efficient and have advantages over some existing ones.
The semiclassical approximation for the space fractional Schrödinger equation (FSE) in high-dimensional spaces is derived, where the phase and amplitude of the wavefunction are proved to be determined by an eikonal equation and a transport equation, respectively. The formulations extend the results of the one-dimensional (1-D) problems studied in Gao et al. (Commun Appl Math Comput, 2024. https://doi.org/10.1007/s42967-024-00384-z). Similarly, these equations reduce to the eikonal and transport equations in the semiclassical approximation for the standard Schrödinger equation with integer-order derivatives as the fractional-order approaches two, and the Hamiltonian is consistent with that in the classical Hamilton-Jacobi approach. High-order Lax-Friedrichs schemes with Runge-Kutta time integration and weighted essentially non-oscillatory finite-difference approximations are adopted to solve the eikonal and transport equations numerically for their solutions such that they can be used to approximate the wavefunction, along with numerical experiments to demonstrate the effectiveness of the semiclassical approximation.
In this paper, a novel augmented Lagrangian preconditioner based on the global Arnoldi for accelerating the convergence of Krylov subspace methods is applied to linear systems of equations with a block three-by-three structure, and these systems typically arise from discretizing the Stokes equations using mixed finite-element methods. Spectral analyses are established for the exact versions of these preconditioners. Finally, the obtained numerical results claim that our novel approach is more efficient and robust for solving the discrete Stokes problems. The efficiency of our new approach is evaluated by measuring the computational time.
In this paper, we study the a posteriori error estimates of the conforming mixed method for the modified transmission eigenvalue problem proposed by Cogar et al. (Inverse Problems 33: 055010, 2017). We give the a posteriori error estimator of the approximate eigenpair, prove the reliability and efficiency of the estimator for the approximate eigenfunction, and present the reliability of the estimator for the approximate eigenvalue. We also implement adaptive computation and exhibit the numerical experiments which show that the approximate eigenvalues obtained by the adaptive computation reach the optimal convergence order.
Mathematical modelling is fundamental to understanding real-world phenomena. Despite the inherent complexity in designing such models, numerical approaches and, more recently, machine learning techniques, have emerged as powerful tools in this area. This work proposes integrating the finite element method (FEM) into forecasting and introduces parallel techniques for regression problems, with a specific focus on the use of Matérn kernels on local mesh support. This approach generalises the modelling based on radial basis function kernels and offers more flexibility to control the smoothness of the modelled functions. An exhaustive study explores the impact of diverse norms and Matérn kernel variations on the performance of models, and aims to improve the computational efficiency of the model fitting and prediction processes. Furthermore, a heuristic framework is introduced to derive optimal complexity parameters for each Matérn-based FEM kernel. The proposed parallel approaches use dynamic strategies, which significantly reduce the computational time of the algorithms compared to other methods and parallel computing techniques presented in recent years. The proposed methodology is assessed in the context of bias corrections for temperature forecasts made by the Local Data Assimilation and Prediction System (LDAPS) model. A comprehensive comparative analysis which includes machine learning algorithms provides significant insights into the training process, norm selection, and kernel choice, and shows that Matérn-based methods emerge as a choice to be considered for regression problems.
The strong vertex (edge) span of a given graph G is the maximum distance that two players can maintain at all times while visiting all vertices (edges) of G and moving either to an adjacent vertex or staying in the current position independently of each other. We introduce the notions of switching walks and the triod size of a tree, which are used to determine the strong vertex and the strong edge span of an arbitrary tree. The obtained results are used in an algorithm that computes the strong vertex (edge) span of the input tree in linear time.
Two-fluid relativistic plasma flow equations combine the equations of relativistic hydrodynamics (RHD) with Maxwell’s equations for electromagnetic fields, which involve divergence constraints for the magnetic and electric fields. When developing numerical schemes for the model, the divergence constraints are ignored, or Maxwell’s equations are reformulated as perfectly hyperbolic Maxwell’s (PHM) equations by introducing additional equations for correction potentials. In the latter case, the divergence constraints are preserved only as the limiting case. In this article, we present second-order numerical schemes that preserve the divergence constraints for electric and magnetic fields at the discrete level. The schemes are based on using a multidimensional Riemann solver at the vertices of the cells to define the numerical fluxes on the edges. The second-order accuracy is obtained by reconstructing the electromagnetic fields at the corners using a MinMod limiter. The discretization of Maxwell’s equations can be combined with any consistent and stable discretization of the fluid parts. In particular, we consider entropy-stable schemes for the fluid part. The resulting schemes are second-order accurate, entropy stable, and preserve the divergence constraints of the electromagnetic fields. We use explicit and Implicit-Explicit-based (IMEX-based) time discretizations. We then test these schemes using several one- and two-dimensional test cases. We also compare the divergence constraint errors of the proposed schemes with schemes having no divergence constraints treatment and schemes based on the PHM-based divergence cleaning.
In this paper, the time fractional mobile/immobile diffusion equation with the weak singular solution at the initial time is studied. The averaged L1 finite difference scheme is established for the equation. The stability of the numerical scheme is analyzed by the Fourier analysis method. The convergence order of the scheme is , where and h are the sizes of the time and space steps, respectively. In addition, due to the historical dependence of the time fractional derivative, we establish a fast method based on the exponential-sum-approximation, effectively reducing computation and storage. Furthermore, we provide an error estimate of the fast algorithm. Finally, a numerical experiment verifies the effectiveness of the algorithm.
We study a greedy coordinate descent method to solve large linear least-squares problems expanding on a randomized coordinate descent method presented by Leventhal and Lewis (Math. Oper. Res. 35: 641–654, 2010). For an overdetermined system, they proved its exponential convergence, regardless of its consistency. In our work, we study a greedy selection rule for the coordinate descent method which we refer to as the two-dimensional maximal residual Gauss-Seidel (D2MRGS) method. In this method, we select two coordinates in every iteration and treat the current approximation in those directions. Convergence is analyzed for the stated method and numerical experiments are provided to demonstrate its efficiency.