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Finding multiple solutions to elliptic systems with polynomial nonlinearity Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200130
Xuping Zhang; Jintao Zhang; Bo YuElliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator

Improved L2 and H1 error estimates for the Hessian discretization method Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200129
Devika ShylajaThe Hessian discretization method (HDM) for fourth‐order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit‐conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators

An enhanced finite difference time domain method for two dimensional Maxwell's equations Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200123
Timothy Meagher; Bin Jiang; Peng JiangAn enhanced finite‐difference time‐domain (FDTD) algorithm is built to solve the transverse electric two‐dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across

An asymptotic preserving scheme on staggered grids for the barotropic Euler system in low Mach regimes Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200123
Thierry Goudon; Julie Llobell; Sebastian MinjeaudWe present a new scheme for the simulation of the barotropic Euler equation in low Mach regimes. The method uses two main ingredients. First, the system is treated with a suitable time splitting strategy, directly inspired from the previous study that separates low and fast waves. Second, we adapt a numerical scheme where the discrete densities and velocities are stored on staggered grids, in the spirit

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200121
Demin Liu; Linjin LiIn this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three‐dimensional Navier–Stokes equations in the cylinder while the asymptotic

High order convergent modified nodal bi‐cubic spline collocation method for elliptic partial differential equation Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200120
Suruchi Singh; Swarn SinghA high order modified nodal bi‐cubic spline collocation method is proposed for numerical solution of second‐order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena

High‐order dual‐parametric finite element methods for cavitation computation in nonlinear elasticity Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200110
Weijie Huang; Weijun Ma; Liang Wei; Zhiping LiIn this paper, we present the numerical analysis on high order dual parametric finite element methods for the cavitation computation problems in nonlinear elasticity, which leads to a meshing strategy assuring high efficiency on numerical approximations to cavity deformations. Furthermore, to cope with the high order approximation of the finite element methods, properly chosen weighted Gaussian type

Energy stability and convergence of the scalar auxiliary variable Fourier‐spectral method for the viscous Cahn–Hilliard equation Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200107
Nan Zheng; Xiaoli LiIn this paper, we develop two linear and unconditionally energy stable Fourier‐spectral schemes for solving viscous Cahn–Hilliard equation based on the recently scalar auxiliary variable approach. The temporal discretizations are built upon the first‐order Euler method and second‐order Crank–Nicolson method, respectively. We carry out the energy stability and error analysis rigorously. Various classical

Numerical analysis of a parabolic variational inequality system modeling biofilm growth at the porescale Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200107
Azhar Alhammali; Malgorzata PeszynskaIn this article, we consider a system of two coupled nonlinear diffusion–reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element approximation

DOF‐gathering stable generalized finite element methods for crack problems Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20200106
Qinghui ZhangGeneralized or eXtended finite element methods (GFEM/XFEM) have been studied extensively for crack problems. Most of the studies were concentrated on localized enrichment schemes where nodes around the crack tip are enriched by products of singular and finite element shape functions. To attain the optimal convergence rate O(h) (h is the mesh‐size), nodes in a fixed domain containing the tip have to

Numerical method for generalized time fractional KdV‐type equation Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191225
Desong Kong; Yufeng Xu; Zhoushun ZhengIn this article, an efficient numerical method for linearized and nonlinear generalized time‐fractional KdV‐type equations is proposed by combining the finite difference scheme and Petrov–Galerkin spectral method. The scale and weight functions involved in generalized fractional derivative cause too much difficulty in discretization and numerical analysis. Fortunately, motivated by finite difference

A parameter‐uniform scheme for singularly perturbed partial differential equations with a time lag Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191224
Devendra Kumar; Parvin KumariA numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter

A finite difference scheme for smooth solutions of the general Degasperis–Procesi equation Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191223
Jesus Noyola Rodriguez; Georgy Omel'yanovThe general Degasperis–Procesi equation (gDP) describes the evolution of the water surface in a unidirectional shallow water approximation. We propose a finite‐difference scheme for this equation that preserves some conservation and balance laws. In addition, the stability of the scheme and the convergence of numerical solutions to exact solutions for solitons are proved. Numerical experiments confirm

A modified sensitivity equation method for the Euler equations in presence of shocks Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191218
Camilla Fiorini; Christophe Chalons; Régis DuvigneauThe continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution

Second‐order, loosely coupled methods for fluid‐poroelastic material interaction Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191213
Oyekola Oyekole; Martina BukačThis work focuses on modeling the interaction between an incompressible, viscous fluid and a poroviscoelastic material. The fluid flow is described using the time‐dependent Stokes equations, and the poroelastic material using the Biot model. The viscoelasticity is incorporated in the equations using a linear Kelvin–Voigt model. We introduce two novel, noniterative, partitioned numerical schemes for

Fourier spectral method with an adaptive time strategy for nonlinear fractional Schrödinger equation Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191211
Haidong Qu; Zihang SheIn this paper, a Fourier spectral method with an adaptive time step strategy is proposed to solve the fractional nonlinear Schrödinger (FNLS) equation with periodic initial value problem. First, we prove the conservation law of the mass and the energy for the semi‐discrete Fourier spectral scheme. Second, the error estimation of the semi‐discrete scheme is given in the relevant fractional Sobolev space

Numerical algorithms for the time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191210
Hengfei Ding; Changpin LiIn this paper, high‐order numerical methods for time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations in one‐ and two‐dimensional space are constructed, where the second‐order backward fractional difference operator and the sixth‐order fractional‐compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the

A discontinuous Galerkin method for Stokes equation by divergence‐free patch reconstruction Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191130
Ruo Li; Zhiyuan Sun; Zhijian YangA discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence‐free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as an elliptic system instead of a saddle‐point problem

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191126
Bhupen Deka; Papri RoyIn this paper, the weak Galerkin finite element method (WG‐FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG‐FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity

Well‐posedness and finite element approximation of time dependent generalized bioconvective flow Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191118
Yanzhao Cao; Song Chen; Hans‐Werner van WykWe consider the finite element approximation of a time dependent generalized bioconvective flow. The underlying system of partial differential equations consists of incompressible Navier–Stokes type convection equations coupled with an equation describing the transport of micro‐organisms. The viscosity of the fluid is assumed to be a function of the concentration of the micro‐organisms. We show the

Numerical analysis of the coupling of free fluid with a poroelastic material Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191106
Aycil Cesmelioglu; Prince ChidyagwaiThis paper presents a numerical solution of the coupled system of the time‐dependent Stokes and fully dynamic Biot equations. The numerical scheme is based on standard inf‐sup stable finite elements in space and the Backward Euler scheme in time. We establish stability of the scheme and derive error estimates for the fully discrete coupled scheme. To handle realistic parameters which may cause nonphysical

High‐order discrete‐time orthogonal spline collocation methods for singularly perturbed 1D parabolic reaction–diffusion problems Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191106
Pankaj Mishra; Kapil K. Sharma; Amiya K. Pani; Graeme FairweatherQuasi‐optimal error estimates are derived for the continuous‐time orthogonal spline collocation (OSC) method and also two discrete‐time OSC methods for approximating the solution of 1D parabolic singularly perturbed reaction–diffusion problems. OSC with C1 splines of degree r ≥ 3 on a Shishkin mesh is employed for the spatial discretization while the Crank–Nicolson method and the BDF2 scheme are considered

A hybrid high‐order formulation for a Neumann problem on polytopal meshes Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191126
Rommel Bustinza; Jonathan Munguia‐La‐CoteraIn this work, we study a hybrid high‐order (HHO) method for an elliptic diffusion problem with Neumann boundary condition. The proposed method has several features, such as: (a) the support of arbitrary approximation order polynomial at mesh elements and faces on polytopal meshes, (b) the design of a local (element‐wise) potential reconstruction operator and a local stabilization term, that weakly

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191118
Yujie Liu; Junping WangThis article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony‐type equation with nonsmooth solutions Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191118
Pin Lyu; Seakweng VongTo recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time‐fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191121
Jintao Cui; Fuzheng Gao; Zhengjia Sun; Peng ZhuIn this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness

A fast algorithm for a total variation based phase demodulation model Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191126
Carlos Brito‐Loeza; Ricardo Legarda‐Saenz; Anabel Martin‐GonzalezIn this paper we introduce fast numerical algorithms for the solution of the model. For each variable, background illumination, amplitude modulation and phase map, we develop a fixed point method. Then, we write all three algorithms in the same framework and analyze their convergence rates, local smoothing factors by means of local Fourier analysis and present experimental evidence of their performance

Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker–Planck equation Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191118
Ali Namadchian; Mehdi RamezaniThe Fokker–Planck equation is a useful tool to analyze the transient probability density function of the states of a stochastic differential equation. In this paper, a multilayer perceptron neural network is utilized to approximate the solution of the Fokker–Planck equation. To use unconstrained optimization in neural network training, a special form of the trial solution is considered to satisfy the

Splitting method for an inverse source problem in parabolic differential equations: Error analysis and applications Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191128
Simin Shekarpaz; Hossein AzariIn this work, we present a numerical method based on a splitting algorithm to find the solution of an inverse source problem with the integral condition. The source term is reconstructed by using the specified data and by employing the Lie splitting method, we decompose the equation into linear and nonlinear parts. Each subproblem is solved by the Fourier transform and then by combining the solutions

Numeric solution of advection–diffusion equations by a discrete time random walk scheme Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20191121
Christopher N. Angstmann; Bruce I. Henry; Byron A. Jacobs; Anna V. McGannExplicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable

Double complexes and local cochain projections. Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20150429
Richard S Falk,Ragnar WintherThe construction of projection operators, which commute with the exterior derivative and at the same time are bounded in the proper Sobolev spaces, represents a key tool in the recent stability analysis of finite element exterior calculus. These socalled bounded cochain projections have been constructed by combining a smoothing operator and the unbounded canonical projections defined by the degrees

Dynamic DataDriven Finite Element Models for Laser Treatment of Cancer. Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : 20070426
J T Oden,K R Diller,C Bajaj,J C Browne,J Hazle,I Babuška,J Bass,L Biduat,L Demkowicz,A Elliott,Y Feng,D Fuentes,S Prudhomme,M N Rylander,R J Stafford,Y ZhangElevating the temperature of cancerous cells is known to increase their susceptibility to subsequent radiation or chemotherapy treatments, and in the case in which a tumor exists as a welldefined region, higher intensity heat sources may be used to ablate the tissue. These facts are the basis for hyperthermia based cancer treatments. Of the many available modalities for delivering the heat source

Finitevolume scheme for a degenerate crossdiffusion model motivated from ion transport. Numer. Methods Partial Differ. Equ. (IF 1.633) Pub Date : null
Clément Cancès,Claire ChainaisHillairet,Anita Gerstenmayer,Ansgar JüngelAn implicit Euler finitevolume scheme for a degenerate crossdiffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The crossdiffusion system possesses a formal gradientflow structure