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A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-03-16 Huadong Gao, Wen Xie
This article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element
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Unconditional optimal first‐order error estimates of a full pressure segregation scheme for the magnetohydrodynamics equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-03-16 Yun‐Bo Yang, Yao‐Lin Jiang
In this article, a first‐order linear fully discrete pressure segregation scheme is studied for the time‐dependent incompressible magnetohydrodynamics (MHD) equations in three‐dimensional bounded domain. Based on an incremental pressure projection method, this scheme allows us to decouple the MHD system into two sub‐problems at each time step, one is the velocity‐magnetic field system, the other is
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Adaptive finite element methods for scalar double‐well problem Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-03-08 Bingzhen Li, Dongjie Liu
Some scalar double‐well problems eventually lead to a degenerate convex minimization problem with unique stress. We consider the adaptive conforming and nonconforming finite element methods for the scalar double‐well problem with the reliable a posteriori error analysis. A number of experiments confirm the effective decay rates of the methods.
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On strong convergence of a fully discrete scheme for solving stochastic strongly damped wave equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-02-17 Chengqiang Xu, Yibo Wang, Wanrong Cao
This article develops an efficient fully discrete scheme for a stochastic strongly damped wave equation (SSDWE) driven by an additive noise and presents its error estimates in the strong sense. We use the truncated spectral expansion of the noise to get an approximate equation and prove its regularity. Then we establish a spatio-temporal discretization of the approximate equation by a finite element
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Discrete null field equation methods for solving Laplace's equation: Boundary layer computations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-02-14 Li-Ping Zhang, Zi-Cai Li, Ming-Gong Lee, Hung-Tsai Huang
Consider Dirichlet problems of Laplace's equation in a bounded simply-connected domain S$$ S $$, and use the null field equation (NFE) of Green's representation formulation, where the source nodes Q$$ Q $$ are located on a pseudo-boundary ΓR$$ {\Gamma}_R $$ outside S$$ S $$ but not close to its boundary Γ(=∂S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$. Simple algorithms are proposed in this
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New quadratic/serendipity finite volume element solutions on arbitrary triangular/quadrilateral meshes Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-02-14 Yanhui Zhou
By postprocessing quadratic and eight-node serendipity finite element solutions on arbitrary triangular and quadrilateral meshes, we obtain new quadratic/serendipity finite volume element solutions for solving anisotropic diffusion equations. The postprocessing procedure is implemented in each element independently, and we only need to solve a 6-by-6 (resp. 8-by-8) local linear algebraic system for
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A posteriori error analysis of a semi-augmented finite element method for double-diffusive natural convection in porous media Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-02-13 Mario Álvarez, Eligio Colmenares, Filánder A. Sequeira
This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi-augmented mixed-primal finite element method previously developed by us to numerically solve double-diffusive natural convection problem in porous media. The model combines Brinkman-Navier-Stokes equations for velocity and pressure coupled to a vector advection-diffusion equation, representing heat and concentration
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A second-order time discretizing block-centered finite difference method for compressible wormhole propagation Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-02-07 Fei Sun, Xiaoli Li, Hongxing Rui
In this paper, a second-order time discretizing block-centered finite difference method is introduced to solve the compressible wormhole propagation. The optimal second-order error estimates for the porosity, pressure, velocity, concentration and its flux are established carefully in different discrete norms on non-uniform grids. Then by introducing Lagrange multiplier, a novel bound-preserving scheme
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Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-01-30 Yali Gao, Daozhi Han
We develop two totally decoupled, linear and second-order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele-Shaw cell. The implicit-explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression
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Retraction: Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-02-01
Retraction: Nurgabyl DN, Uaissov AB. Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps. Numer Methods Partial Differential Eq. 2021; 37: 2375–2392. https://doi.org/10.1002/num.22719
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On the convergence of 2D P4+ triangular and 3D P6+ tetrahedral divergence-free finite elements Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-01-21 Shangyou Zhang
We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the P k $$ {P}_k $$ - P k − 1 disc $$ {P}_{k-1}^{\mathrm{disc}} $$ mixed finite element method for k ≥ 4 $$ k\ge 4 $$ on 2D triangular grids or k ≥ 6 $$ k\ge 6 $$ on tetrahedral grids, even in the case the inf-sup condition fails. By a simple L 2 $$ {L}^2 $$ -projection of the
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Double diffusive effects on nanofluid flow toward a permeable stretching surface in presence of Thermophoresis and Brownian motion effects: A numerical study Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2024-01-19 V. V. L. Deepthi, V. K. Narla, R. Srinivasa Raju
The present study explores the nanofluid boundary layer flow over a stretching sheet with the combined influence of the double diffusive effects of thermophoresis and Brownian motion effects. For the purpose of transforming nonlinear partial differential equations into the linear united ordinary differential equation method, the similarity transformation technique is used. The Runge–Kutta–Fehlberg
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Improved error estimates of the time-splitting methods for the long-time dynamics of the Klein–Gordon–Dirac system with the small coupling constant Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-12-11 Jiyong Li
We provide improved uniform error estimates for the time-splitting Fourier pseudo-spectral (TSFP) methods applied to the Klein–Gordon–Dirac system (KGDS) with the small parameter ◂+▸ε∈(0,1]$$ \varepsilon \in \left(0,1\right] $$. We first reformulate the KGDS into a coupled Schrödinger–Dirac system (CSDS) and then apply the second-order Strang splitting method to CSDS with the spatial discretization
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Iteration acceleration methods for solving three-temperature heat conduction equations on distorted meshes Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-11-27 Yunlong Yu, Xingding Chen, Yanzhong Yao
This article focuses on designing efficient iteration algorithms for nonequilibrium three-temperature heat conduction equations, which are used to formulate the radiative energy transport problem. Based on the framework of relaxation iteration, we design a new accelerated iteration algorithm by reasonable approximation of the Jacobi matrix, according to the characteristics of the discrete scheme for
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Robust finite element methods and solvers for the Biot–Brinkman equations in vorticity form Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-11-27 Ruben Caraballo, Chansophea Wathanak In, Alberto F. Martín, Ricardo Ruiz-Baier
In this article, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis
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A radial basis function (RBF)-finite difference method for solving improved Boussinesq model with error estimation and description of solitary waves Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-11-15 Mostafa Abbaszadeh, AliReza Bagheri Salec, Taghreed Abdul-Kareem Hatim Aal-Ezirej
The Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi-discrete scheme according to the fixed point theorem. In addition, the unconditional
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Efficient and accurate temporal second-order numerical methods for multidimensional multi-term integrodifferential equations with the Abel kernels Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-11-14 Mingchao Zhao, Hao Chen, Kexin Li
This work develops two temporal second-order alternating direction implicit (ADI) numerical schemes for solving multidimensional parabolic-type integrodifferential equations with multi-term weakly singular Abel kernels. For the two-dimensional (2D) case, applying the Crank–Nicolson method and product integration rule to discretizations of temporal derivative and integral terms, respectively, and the
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Unfitted generalized finite element methods for Dirichlet problems without penalty or stabilization Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-11-09 Qinghui Zhang
Unfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM)
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Numerical approximation for hybrid-dimensional flow and transport in fractured porous media Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-11-05 Jijing Zhao, Hongxing Rui
This article presents the stable miscible displacement problem in fractured porous media, and finite element discretization is constructed for this reduced model. The transmission interface conditions presented in this article enable us to derive a stability result and conduct the case where the pressure and concentration are both discontinuous across the fracture. The error estimates for H1$$ {H}^1
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Numerical algorithm with fifth-order accuracy for axisymmetric Laplace equation with linear boundary value problem Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-10-30 Hu Li, Jin Huang
In order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid-rectangle formula with a
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Error analyses on block-centered finite difference schemes for distributed-order non-Fickian flow Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-10-24 Xuan Zhao, Ziyan Li
In this article, two numerical schemes are designed and analyzed for the distributed-order non-Fickian flow. Two different processing techniques are applied to deal with the time distributed-order derivative for the constructed two schemes, while the classical block-centered finite difference method is used in spatial discretization. To be precise, one adopts the standard numerical scheme called SD
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Richardson extrapolation method for solving the Riesz space fractional diffusion problem Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-10-19 Ren-jun Qi, Zhi-zhong Sun
The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the
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A semi-Lagrangian ε-monotone Fourier method for continuous withdrawal GMWBs under jump-diffusion with stochastic interest rate Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-10-19 Yaowen Lu, Duy-Minh Dang
We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump-diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no-arbitrage GMWB pricing problem as a time-dependent Hamilton-Jacobi-Bellman (HJB) Quasi-Variational Inequality (QVI) having three
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Implicit Runge-Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-10-12 Yanming Zhang, Yu Li, Yuexin Yu, Wansheng Wang
An efficient numerical method with high accuracy both in time and in space is proposed for solving d$$ d $$-dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an s$$ s $$-stage implicit Runge-Kutta method and approximating the space by a spectral Galerkin method with Fourier-like basis functions. In view of the orthogonality, the
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Two-grid finite element method on grade meshes for time-fractional nonlinear Schrödinger equation Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-10-09 Hanzhang Hu, Yanping Chen, Jianwei Zhou
A two-grid finite element method with nonuniform L1 scheme is developed for solving the time-fractional nonlinear Schrödinger equation. The finite element solution in the L p $$ {L}^p $$ -norm and L ∞ $$ {L}^{\infty } $$ -norm are proved bounded without any time-step size conditions (dependent on spatial-step size). Then, the optimal order error estimations of the two-grid solution in the L p $$ {L}^p
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A Legendre spectral-Galerkin method for fourth-order problems in cylindrical regions Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-09-22 Jihui Zheng, Jing An
A spectral-Galerkin method based on Legendre-Fourier approximation for fourth-order problems in cylindrical regions is studied in this paper. By the cylindrical coordinate transformation, a three-dimensional fourth-order problem in a cylindrical region is transformed into a sequence of decoupled fourth-order problems with two dimensions and the corresponding pole conditions are also derived. With appropriately
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Analysis of two fully discrete spectral volume schemes for hyperbolic equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-09-20 Ping Wei, Qingsong Zou
In this paper, we analyze two classes of fully discrete spectral volume schemes (SV) for solving the one-dimensional scalar hyperbolic equation. These two schemes are constructed by using the forward Euler (EU) method or the second-order Runge–Kutta (RK2) method in time-discretization, and by letting a piecewise kth degree( k ≥ 1 $$ k\ge 1 $$ is an arbitrary integer) polynomial satisfy the local conservation
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Numerical study of conforming space-time methods for Maxwell's equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-09-14 Julia I. M. Hauser, Marco Zank
Time-dependent Maxwell's equations govern electromagnetics. Under certain conditions, we can rewrite these equations into a partial differential equation of second order, which in this case is the vectorial wave equation. For the vectorial wave equation, we examine numerical schemes and their challenges. For this purpose, we consider a space-time variational setting, that is, time is just another spatial
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Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-09-03 Ye Hu, Yubin Yan, Shahzad Sarwar
Recently, Kovács et al. considered a Mittag-Leffler Euler integrator for a stochastic semilinear Volterra integral-differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66-85]. In this article, we shall consider the Mittag-Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion
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Analysis and application of a local discontinuous Galerkin method for the electromagnetic concentrator model Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-09-01 Yunqing Huang, Jichun Li, Xin Liu
In this paper, we develop a local discontinuous Galerkin (LDG) method to simulate the wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proved for both semi-discrete and full-discrete LDG schemes. Numerical results are presented
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Finite element algorithm with a second-order shifted composite numerical integral formula for a nonlinear time fractional wave equation Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-08-30 Haoran Ren, Yang Liu, Baoli Yin, Hong Li
In this article, we propose a second-order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical
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Stability and temporal error estimate of scalar auxiliary variable schemes for the magnetohydrodynamics equations with variable density Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-08-28 Hang Chen, Yuyu He, Hongtao Chen
In this article, we construct first- and second-order semidiscrete schemes for the magnetohydrodynamics (MHD) equations with variable density based on scalar auxiliary variable (SAV) approach. These schemes are decoupled, unconditionally energy stable and only solve a sequence of linear differential equations at each time step. We carry out a rigorous error analysis for the first-order SAV scheme in
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A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-08-14 Jehanzeb H. Chaudhry, Donald Estep, Simon J. Tavener
We construct a space-time parallel method for solving parabolic partial differential equations by coupling the parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of “local” problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating
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Unfitted mixed finite element methods for elliptic interface problems Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-08-11 Najwa Alshehri, Daniele Boffi, Lucia Gastaldi
In this article, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid-structure interaction problems. Two finite element schemes with piecewise constant Lagrange multiplier
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An efficient linearly implicit and energy-conservative scheme for two dimensional Klein–Gordon–Schrödinger equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-08-04 Hongwei Li, Yuna Yang, Xiangkun Li
The Klein–Gordon–Schrödinger equations describe a classical model of interaction of nucleon field with meson field in physics, how to design the energy conservative and stable schemes is an important issue. This paper aims to develop a linearized energy-preserve, unconditionally stable and efficient scheme for Klein–Gordon–Schrödinger equations. Some auxiliary variables are utilized to circumvent the
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Strong convergence for an explicit fully-discrete finite element approximation of the Cahn-Hillard-Cook equation with additive noise Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-07-28 Qiu Lin, Ruisheng Qi
In this paper, we consider an explicit fully-discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity-tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform
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Spatio-temporal scalar auxiliary variable approach for the nonlinear convection–diffusion equation with discontinuous Galerkin method Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-07-19 Yaping Li, Weidong Zhao, Wenju Zhao
In this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient-type nonlinear term. The nonlinear convection–diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable
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Superconvergence analysis of the bilinear-constant scheme for two-dimensional incompressible convective Brinkman–Forchheimer equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-07-18 Huaijun Yang, Xu Jia
In this article, a low order conforming mixed finite element method is proposed and investigated for two-dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear-constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in H 1 $$ {H}^1 $$ -norm and pressure in L 2 $$
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Two-sided Krylov enhanced proper orthogonal decomposition methods for partial differential equations with variable coefficients Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-07-05 Li Wang, Zhen Miao, Yao-Lin Jiang
In this paper, new fast computing methods for partial differential equations with variable coefficients are studied and analyzed. They are two kinds of two-sided Krylov enhanced proper orthogonal decomposition (KPOD) methods. First, the spatial discrete scheme of an advection-diffusion equation is obtained by Galerkin approximation. Then, an algorithm based on a two-sided KPOD approach involving the
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A priori error estimates of two monolithic schemes for Biot's consolidation model Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-29 Huipeng Gu, Mingchao Cai, Jingzhi Li, Guoliang Ju
This paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three-field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain
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Neilan's divergence-free finite elements for Stokes equations on tetrahedral grids Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-28 Shangyou Zhang
The Neilan P k $$ {P}_k $$ - P k − 1 $$ {P}_{k-1} $$ divergence-free finite element is stable on any tetrahedral grid, where the piece-wise P k $$ {P}_k $$ polynomial velocity is C 0 $$ {C}^0 $$ on the grid, C 1 $$ {C}^1 $$ on edges and C 2 $$ {C}^2 $$ at vertices, and the piece-wise P k − 1 $$ {P}_{k-1} $$ polynomial pressure is C 0 $$ {C}^0 $$ on edges and C 1 $$ {C}^1 $$ at vertices. However the
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Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-26 Qifeng Zhang, Jiyuan Zhang, Zhi-zhong Sun
This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high-dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal
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An efficient and accurate numerical method for the fractional optimal control problems with fractional Laplacian and state constraint Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-26 Jiaqi Zhang, Yin Yang
In this paper, we investigate the numerical approximation of an optimal control problem with fractional Laplacian and state constraint in integral form based on the Caffarelli–Silvestre expansion. The first order optimality conditions of the extended optimal control problem is obtained. An enriched spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is
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Conservative EQ1rot nonconforming FEM for nonlinear Schrödinger equation with wave operator Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-26 Lingli Wang, Meng Li, Shanshan Peng
In this paper, we consider leap-frog finite element methods with EQ 1 rot $$ {\mathrm{EQ}}_1^{\mathrm{rot}} $$ element for the nonlinear Schrödinger equation with wave operator. We propose that both the continuous and discrete systems can keep mass and energy conservation. In addition, we focus on the unconditional superconvergence analysis of the numerical scheme, the key of which is the time-space
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A new high order hybrid WENO scheme for hyperbolic conservation laws Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-19 Liang Li, Zhenming Wang, Zhong Zhao, Jun Zhu
This article proposes an improved hybrid weighted essentially non-oscillatory (WENO) scheme based on the third- and fifth-order finite-difference modified WENO (MWENO) schemes developed by Zhu et al. in (SIAM J. Sci. Comput. 39 (2017), A1089–A1113.) for solving hyperbolic conservation laws. The MWENO schemes give a guideline on whether to use the WENO scheme or the linear upwind scheme. Unfortunately
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Experimental convergence rate study for three shock-capturing schemes and development of highly accurate combined schemes Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-14 Shaoshuai Chu, Olyana A. Kovyrkina, Alexander Kurganov, Vladimir V. Ostapenko
We study experimental convergence rates of three shock-capturing schemes for hyperbolic systems of conservation laws: the second-order central-upwind (CU) scheme, the third-order Rusanov-Burstein-Mirin (RBM), and the fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and W − 1 , 1 $$ {W}^{-1,1} $$
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Behavior of Lagrange-Galerkin solutions to the Navier-Stokes problem for small time increment Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-06-04 Masahisa Tabata, Shinya Uchiumi
We consider two kinds of numerical quadrature formulas of Gauss type and Newton-Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment
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A finite volume scheme preserving the invariant region property for a class of semilinear parabolic equations on distorted meshes Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-26 Huifang Zhou, Yuanyuan Liu, Zhiqiang Sheng
In this article, we present a finite volume scheme preserving invariant-region-property (IRP) for a class of semilinear parabolic equations with anisotropic diffusion coefficient on distorted meshes. The diffusion term is discretized by the finite volume scheme preserving the discrete maximum principle, and the time derivative is discretized by the backward Euler scheme. For the nonlinear system, a
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A mass- and energy-preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-24 Jianfeng Liu, Qinglin Tang, Tingchun Wang
This article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin-orbit-coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled
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Virtual element method for elliptic bulk-surface PDEs in three space dimensions Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-22 Massimo Frittelli, Anotida Madzvamuse, Ivonne Sgura
In this work we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk-surface partial differential equations in three space dimensions. The BSVEM is based on the discretization of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. We present
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Two-level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-20 Xiaochen Chu, Chuanjun Chen, Tong Zhang
In this paper, a two-level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf-sup condition. Firstly, the existence and uniqueness of the solution
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A linear adaptive second-order backward differentiation formulation scheme for the phase field crystal equation Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-17 Dianming Hou, Zhonghua Qiao
In this article, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed
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Parameter robust higher-order finite difference method for convection-diffusion problem with time delay Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-16 Sanjay Ku Sahoo, Vikas Gupta
This paper deals with the study of a higher-order numerical approximation for a class of singularly perturbed convection-diffusion problems with time delay. The method combines a higher-Order Difference with an Identity Expansion (HODIE) scheme over a piece-wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori
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A simplified primal-dual weak Galerkin finite element method for Fokker–Planck-type equations Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-10 Dan Li, Chunmei Wang
A simplified primal-dual weak Galerkin (S-PDWG) finite element method is designed for the Fokker–Planck-type equation with nonsmooth diffusion tensor and drift vector. The discrete system resulting from S-PDWG method has significantly fewer degrees of freedom compared with the one resulting from the PDWG method proposed by Wang-Wang. Furthermore, the condition number of the S-PDWG method is smaller
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Nitsche's method for a Robin boundary value problem in a smooth domain Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-10 Yuki Chiba, Norikazu Saito
We prove several optimal-order error estimates for a 𝒫 1 finite-element method applied to an inhomogeneous Robin boundary value problem (BVP) for the Poisson equation defined in a smooth bounded domain in ℝ n $$ {\mathbb{R}}^n $$ , n = 2 , 3 $$ n=2,3 $$ . The boundary condition is weakly imposed using Nitsche's method. The Robin BVP is interpreted as the classical penalty method with the penalty parameter
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High order accurate and convergent numerical scheme for the strongly anisotropic Cahn–Hilliard model Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-04 Kelong Cheng, Cheng Wang, Steven M. Wise
We propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well-posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second
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A positivity preserving high-order finite difference method for compressible two-fluid flows Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-05-04 Daniel Boe, Khosro Shahbazi
Any robust computational scheme for compressible flows must retain the hyperbolicity property or the real-valued sound speed. Failure to maintain hyperbolicity, or the positivity of the square of the speed of sound, causes nonphysical distortions and the blow-up of numerical simulations. Strong shock waves and interfacial discontinuities are ubiquitous features of the two-fluid compressible dynamics
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A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-04-29 Da Li, Keran Li, Wenyuan Liao
In this paper, we consider the development and analysis of a new explicit compact high-order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth-order accuracy in space and second-order accuracy
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P1$$ {P}_1 $$–Nonconforming quadrilateral finite element space with periodic boundary conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis Numer. Methods Partial Differ. Equ. (IF 3.9) Pub Date : 2023-04-29 Jaeryun Yim, Dongwoo Sheen
The P 1 $$ {P}_1 $$ –nonconforming quadrilateral finite element space with periodic boundary conditions is investigated. The dimension and basis for the space are characterized by using the concept of minimally essential discrete boundary conditions. We show that the situation is different based on the parity of the number of discretizations on coordinates. Based on the analysis on the space, we propose