Advances in Computational Mathematics最新文献

A difference finite element method based on the mixed finite element pair (((P_1^b,P_1^b,P_1) times (P_1,P_1,P_1)))-((P_1 times P_0)) is presented for the three-dimensional stationary Navier–Stokes equations with damping. Moreover, based on this proposed method, a two-level discretization is constructed, which involves solving a problem of the Navier–Stokes equations with damping on coarse mesh with mesh sizes H and (mathcal {T}), and a general Stokes problem on fine mesh with mesh sizes (h = O(H^2)) and (tau = O(mathcal {T}^2)). This two-level difference finite element method provides an approximate solution with the same convergence rate as the difference finite element solution, which involves solving a problem of the Navier–Stokes equations with damping on fine mesh with mesh sizes h and (tau ). Hence, it can save a large amount of computational time. Finally, all computational results support the theoretical analysis and show the effectiveness of the two-level difference finite element method for solving the considered problem.

Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, while the elastic body is assumed to be isotropic and linear. By introducing the Dirichlet-to-Neumann (DtN) operators for acoustic and elastic waves simultaneously, the model is formulated as an acoustic-elastic interaction problem in periodic structures. Based on a duality argument, an a posteriori error estimate is derived for the associated truncated finite element approximation. The a posteriori error estimate consists of the finite element approximation error and the truncation error of two different DtN operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem in periodic structures. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithm.

This study investigates a finite difference numerical scheme to analyze the impact of the effects caused by the strong coupling of Fourier’s law on the solutions of the equations of motion of a mixture of two one-dimensional linear isotropic elastic materials with frictional damping. We first prove the existence of solutions and exponential stability. In the sequence, we analyze the semi-discrete problem in finite differences and we use the energy method to prove the exponential stabilization of the corresponding semi-discrete system. The positivity of the numerical energy is also proved, and we present a fully discrete finite difference scheme that combines explicit and implicit integration methods. Finally, numerical simulations are given to confirm the theoretical results and show the efficiency of the proposed scheme.

We propose a nested Picard iterative integrator Fourier pseudo-spectral (NPI-FP) method and establish the uniform error bounds for the Klein-Gordon-Zakharov system (KGZS) with (varepsilon in (0, 1]) being a small parameter. In the subsonic limit regime ((0 < varepsilon ll 1)), the solution of KGZS propagates waves with wavelength (O(varepsilon )) in time and amplitude at (O(varepsilon ^{alpha ^dagger } )) with (alpha ^dagger =min {alpha ,beta +1,2}), where (alpha ) and (beta ) describe the incompatibility between the initial data of the KGZS and the limiting equation as (varepsilon rightarrow 0^+) and satisfy (alpha ge 0), (beta +1ge 0). The oscillation in time becomes the main difficulty in constructing numerical schemes and making the corresponding error analysis for KGZS in this regime. In this paper, firstly, in order to overcome the difficulty of controlling nonlinear terms, we transform the KGZS into a system with higher derivative. Using the technique of nested Picard iteration, we construct a new time semi-discretization scheme and obtain the error estimates of semi-discretization with the bounds at (O(min {tau ,tau ^2/varepsilon ^{1-alpha ^*}})) for (beta ge 0) where (alpha ^*=min {1,alpha ,1+beta }) and (tau ) is time step. Hence, we get uniformly second-order error bounds at (O(tau ^{2})) when (alpha ge 1) and (beta ge 0), and uniformly accurate first-order error estimates for any (alpha ge 0) and (beta ge 0). We also give full discretization by Fourier pseudo-spectral method and obtain the error bounds at (O(h^{sigma +2}+min {tau ,tau ^2/varepsilon ^{1-alpha ^*}})), where h is mesh size and (sigma ) depends on the regularity of the solution. Hence, we get uniformly accurate spatial spectral order for any (alpha ge 0) and (beta ge 0). Our numerical results support the error estimates. Surprisingly, our numerical results suggest a better error bound at (O(h^{sigma +2}+ varepsilon ^qtau ^{2})) for a certain (qin mathbb {R}).

We investigate reduced order models for acoustic and electromagnetic wave problems in parametrically defined domains. The parameter-to-solution maps are approximated following the so-called Galerkin POD-NN method, which combines the construction of a reduced basis via proper orthogonal decomposition (POD) with neural networks (NNs). As opposed to the standard reduced basis method, this approach allows for the swift and efficient evaluation of reduced order solutions for any given parametric input. As is customary in the analysis of problems in random or parametrically defined domains, we start by transporting the formulation to a reference domain. This yields a parameter-dependent variational problem set on parameter-independent functional spaces. In particular, we consider affine-parametric domain transformations characterized by a high-dimensional, possibly countably infinite, parametric input. To keep the number of evaluations of the high-fidelity solutions manageable, we propose using low-discrepancy sequences to sample the parameter space efficiently. Then, we train an NN to learn the coefficients in the reduced representation. This approach completely decouples the offline and online stages of the reduced basis paradigm. Numerical results for the three-dimensional Helmholtz and Maxwell equations confirm the method’s accuracy up to a certain barrier and show significant gains in online speed-up compared to the traditional Galerkin POD method.

In this paper, we aim to design two energy-stable and efficient finite element schemes for simulating the ferrofluid flows based on the well-known Shliomis model. The model is a highly nonlinear, coupled, multi-physics system, consisting of the Navier–Stokes equations, magnetostatic equation, and magnetization field equation. We propose two reliable numerical algorithms with the following desired features: linearity and unconditional energy stability. Several key techniques are used to achieve the required features, including the auxiliary variable method, consistent terms method, prediction-correction method, and semi-implicit stabilization method. The first scheme is based on a hybrid continuous/discontinuous finite elements spatial approximation, and the second utilizes decoupled continuous finite element spatial discretization. We have rigorously demonstrated that the proposed schemes are unconditionally energy stable and carried out extensive numerical simulations to illustrate the accuracy and stability of the developed schemes, as well as some interesting controllable characteristics of the ferrofluid flows.

Mean value coordinates can be used to map one polygon into another, with application to computer graphics and curve and surface modelling. In this paper, we show that if the polygons are quadrilaterals, and if the target quadrilateral is convex, then the mapping is injective.

In this paper, we investigate a stabilization technique for the Navier–Stokes equations for incompressible fluid flow using equal-order virtual element pairs on general polygonal meshes. We propose a residual-based SUPG-like stabilization term to address the violation of the discrete inf-sup condition, which leads to pressure instability, and to mitigate the effects of the convection-dominated regime. Additionally, we employ a grad-div stabilization term to address the violation of divergence-free constraints. We extend the concept of nonlinear stability derived in (López-Marcos and Sanz-Serna, IMA J. Numer. Anal. 8(1), 71–84, 1998) to a stabilized virtual element framework. Following the results of Lopez-Marcos & Sanz-Serna, we establish the well-posedness and optimal convergence estimates in the energy norm using the branches of non-singular solutions. We perform several numerical experiments to validate the theoretical findings.

The nonconforming virtual element method with curved edges was proposed and analyzed for the Poisson equation by L. Beirão da Veiga, Y. Liu, L. Mascotto, and A. Russo in (J. Sci. Comput. 99(1) 2024). The goal of this paper is to extend the nonconforming virtual element method to a more general second-order elliptic problem with variable coefficients in domains with curved boundaries and curved internal interfaces. We prove an optimal convergence of arbitrary order in the energy and (L^2)-norms, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the method is shown to be comparable with that obtained from the theoretical analysis.

In this paper, based on density functional theory, we present an orthonormal gradient flow (OGF) for finding the ground state solution of a two-dimensional dipolar fermion gas. The OGF has the properties of orthonormality preserving and energy diminishing. By evolving such OGF, we may get the ground state solution of the dipolar fermion gas numerically. The OGF consists of time-dependent integral and partial differential equations. In principle, it can be discretized with many kinds of numerical techniques. We propose a backward Euler Fourier spectral method to discretize such OGF numerically. Numerical tests are reported to demonstrate the effectiveness of the proposed methods. The proposed numerical methods are applied to compute the ground state solution of the ultracold dipolar fermion gas.