This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares

A novel family of isoparametric bilinear finite volume element schemes are constructed and analyzed to solve the anisotropic diffusion problems on general convex quadrilateral meshes. These new schemes are obtained by employing a special quadrature rule to approximate the line integrals in classical -finite volume element method. The new quadrature rule is a linear combination of trapezoidal and midpoint

It has been demonstrated that the MUltiple SIgnal Classification (MUSIC) algorithm is fast, stable, and effective for localizing small anomalies in microwave imaging. For the successful application of MUSIC, exact values of permittivity, conductivity, and permeability of the background must be known. If one of these values is unknown, it will fail to identify the location of an anomaly. However, to

In this work we present two new numerical schemes to approximate the Navier-Stokes-Cahn-Hilliard system with degenerate mobility using finite differences in time and finite elements in space. The proposed schemes are conservative, energy-stable and preserve the maximum principle approximately (the amount of the phase variable being outside of the interval goes to zero in terms of a truncation parameter)

We investigate the temporal stability analysis of a two-layer flow inside a channel that is driven by pressure. The channel consists of a fluid layer overlying an inhomogeneous and anisotropic porous layer. The flow contains a Couette component due to the movement of the horizontal impermeable upper and lower walls binding the two layers. These walls of the channel move at an identical speed but in

Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and convergence. Multigrid is an approach with excellent applicability to elliptic problems such as the Stokes equations, and can be a solution to such challenges of scalability

The gradient discretisation method (GDM) – a generic framework encompassing many numerical methods – is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorokhod theorem and the martingale representation theorem. The generic convergence results established

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two -order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order . The stability of the temporal discretization

Grid-Nesting is a common method of local refinement when using structured quadrilateral grid in estuarine models. Nevertheless, various issues need to be improved, such as the Courant-Friedrichs-Lewy (CFL) limitations of external gravity wave and information exchange between two-way nesting grids. Based on the material conservation law, a novel scheme with Implicit, Grid-Nesting Elevation Solution

This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative

This paper is concerned about the superconvergence analysis for time-dependent Maxwell's equations solved by arbitrary order and basis functions on rectangular and cuboid elements. One-order higher in spatial convergence is proved for leap-frog finite element schemes developed for solving both Maxwell's equations and perfectly matched layer (PML) models. Numerical results for the 2-D PML model solved

For use with the lattice Boltzmann method, the macroscopic boundary conditions need to be transformed into their mesoscopic counterparts. Commonly used mesoscopic boundary conditions use the equilibrium density function, which introduces undesirable artifacts into the numerical solution, especially near interfaces with other types of boundary conditions. In this work, several variants of the mesoscopic

Non-Newtonian fluids within heterogeneous porous media may give rise to complex spatial energy and mass distributions owing to non-local mechanisms, the modeling of which remains unclear. This study investigates the natural convection heat and mass transfer of non-Newtonian fluids in porous media, considering the Soret and Dufour effects. A strongly coupled model is developed to quantify the coupled

In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly reduce the computational load and storage requirements associated with matrix inversion without losing accuracy. To demonstrate the versatility and applicability

In this paper, a residual-type error estimator is proposed and analyzed for a weak Galerkin finite element method for solving linear elasticity problems. The error estimator is proven to be both reliable and efficient, and be used for adaptive refinement. Numerical experiments are presented to illustrate the effectiveness of this error estimator.

Existing nonlocal diffusion models are mainly classified into two categories: bond-based models, which involve a single-fold integral and usually simulate isotropic diffusion, and state-based models, which contain a double-fold integral and can additionally prototype anisotropic diffusion. While bond-based models exhibit more computational efficiency, they sometimes could be limited in modeling capabilities

We consider the reaction-diffusion neural network for which coefficients and neural function depend on time and spatial variable. We study fixed-time anti-synchronization (FTAS) problem. We develop a dual dynamic programming theory to derive verification theorem allowing to find and verify the best fixed-time for anti-synchronization of the system.

In this paper, the effect of random permeability is considered for the real fracture reservoir, hence a stochastic dual-porosity-Navier-Stokes model with random permeability is proposed to simulate shale oil problem based on superhydrophobic material modification. Finite element method and Monte Carlo method are used to deal with discrete physical space and probability space, respectively. The decoupled

In this paper, we develop the Hermite-type virtual element method to solve the second order problem. A Hermite-type virtual element of degree ≥3 is constructed, which can be taken as an extension of classical Hermite finite element to polygonal meshes. For this virtual element, we rigorously prove some inverse inequalities and the boundedness of basis functions. Further, we prove the interpolation

In this paper, the error estimates of the Crank-Nicolson-Leapfrog (CNLF) time-stepping scheme for the two-phase Cahn-Hilliard-Navier-Stokes (CHNS) incompressible flow equations based on scalar auxiliary variable (SAV) are strictly proved. Due to the complexity of the multiple variables and the strong coupling of the equations, it is not easy to prove rigorous error estimates. Under the corresponding

In this article, the mathematical study of the problem of identifying the space-dependent source term, in transport processes given by an -dimensional linear parabolic equation, from space-dependent noisy measurements taken at an arbitrary fixed time is conducted. The problem is analytically solved using Fourier techniques, and it is shown that this solution is not stable. Three families of uniparametric

A nonlinear sea-ice problem is considered in a least-squares finite element setting. The corresponding variational formulation approximating simultaneously the stress tensor and the velocity is analyzed. Under additional smoothness assumptions, the least-squares functional is equivalent to the norm in a neighbourhood of the solution. As the method does not require a compatibility condition between

This paper derives well-posedness and asymptotic results that provide qualitative information about the behavior, mechanism and strategies used by living organisms to navigate their biological networks. Chemical driven swimming is a captivating phenomenon that is observed in various living organisms like bacteria and protozoa but the problem in weighted networks is more complex, since the equations

The finite-difference time-domain (FDTD) method is a widely used numerical technique for simulating electromagnetic wave interactions with complex media. Various efficient approaches have been used to analyze complex media, and the surface impedance boundary condition (SIBC) is one of the most powerful techniques in FDTD simulations, allowing efficient electromagnetic modeling of lossy materials. However

In this study, we propose a novel phase field smoothing-physics informed neural network (PFS-PINN) approach to efficiently solve partial differential equations (PDEs) with discontinuous coefficients. This method combines the phase field model and the PINN model to overcome the difficulty of low regularity solutions and eliminate the limitations of interface constraints in existing neural network solvers

In this paper, we investigate the fractional magnetohydrodynamic (MHD) flow model of a generalized second-grade fluid through a porous medium with Hall current. The fully discrete numerical scheme for solving the model is developed using the formula and Legendre spectral method in time and space, respectively. On the basis of sum-of-exponentials (SOE) technique, a fast formula for the Caputo fractional

The paper introduces an adaptive version of the stabilized Trace Finite Element Method (TraceFEM) designed to solve low-regularity elliptic problems on level-set surfaces using a shape-regular bulk mesh in the embedding space. Two stabilization variants, gradient-jump face and normal-gradient volume, are considered for continuous trace spaces of the first and second degrees, based on the polynomial

We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the boundary value of the solution of the first Laplacian problem. This technique yields a global linear problem that can be solved iteratively via a Krylov-type

This paper is devoted to the study of nonlinear Ginzburg-Landau equation (GLE) with the nonconforming modified quasi-Wilson quadrilateral finite element. Based on the special property of this element, that is its consistency error can reach order in the broken -norm when the exact solution belongs to , and by use of the interpolated postprocessing technique, the superclose and superconvergence estimates

In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh size . While the ratio decreases as for , numerical results indicate a decay of for some in the polynomial

In this work, a finite element-meshless coupling method for modeling the inhomogeneous structures composed of functionally graded materials is presented. Coupling the two methods is usually based on the continuity and equilibrium conditions at the finite element-meshless interface. In the proposed hybrid method, the equilibrium condition is satisfied by the action-reaction principle to ensure the coupling

Forward–backward doubly stochastic differential equations (FBDSDEs) serve as a probabilistic interpretation of stochastic partial differential equations (SPDEs) with diverse applications. Coupled FBDSDEs encounter numerous challenges in numerical approximation compared to forward–backward stochastic differential equations (FBSDEs) and decoupled FBDSDEs, including ensuring the measurability of the numerical

An enriched hybrid high-order method is designed for the Stokes equations of fluid flow and is fully applicable to generic curved meshes. Minimal regularity requirements of the enrichment spaces are given, and an abstract error analysis of the scheme is provided. The method achieves consistency in the enrichment space and is proven to converge optimally in energy error. The scheme is applied to 2D

With their extraordinary ability to conduct heat and their promise to increase heat transfer efficiency, nanofluids have emerged as a major player in the field of fluid technology today. This manuscript delves into the dynamic behavior of time-dependent complex Oldroyd-B nanofluids as they traverse between parallel plates within a porous media. Intriguingly, the study introduces captivating elements

This paper considers a new stabilized finite element method (FEM) of the Stokes equations based on Clément interpolation by the constrained quadrilateral nonconforming rotated - finite element pair. The stabilized term constructed in this method is quite different from those of the existing literature. This method not only has the same attractive computational properties as the conforming stabilized

This paper investigates an induction heating problem in a multi-component system containing a moving non-magnetic conductor. The electromagnetic process is described by the eddy current model, and the heat transfer process is governed by the convection-diffusion equation. The two processes are coupled by a restrained Joule heat source. A temporal discretization scheme is introduced to numerically solve

In this paper, we consider the variable-order time fractional mobile/immobile diffusion (TF-MID) equation in two-dimensional spatial domain, where the fractional order satisfies . We combine the quadratic spline collocation (QSC) method and the formula to propose a QSC- scheme. It can be proved that, the QSC- scheme is unconditionally stable and convergent with , where , Δ and Δ are the temporal and

We identify storage coefficient and hydraulic transmissivity in groundwater flow governed by a linear parabolic equation. Both parameters are assumed to be piecewise constant functions in space. The unknowns are the coefficient values as well as the geometry of the zones where these coefficients are constant. The goal of this work is to improve an adaptive parameterization approach for solving an inverse

In this work, we study the locking-free numerical method for a Barenblatt-Biot poroelastic model. When solving by the continuous Galerkin mixed finite element method, the model exists two kind of locking phenomena for special physical parameters. To overcome these locking phenomena, we introduce new variables to reformulate the original problem into a new problem, which exists a built-in mechanism

In this piece of research, cubic and quartic Hyperbolic B-splines based Differential quadrature methods are implemented for numerical approximation of coupled 2 and 3 Navier-Stokes equations. The validity of the proposed regimes is tested by the means of different variety of errors such as; error, error, error, and error. It is noticed that most of the time, errors generated by cubic Hyperbolic B-spline

We propose a multiple-relaxation-time lattice Boltzmann method for anisotropic convection-diffusion equation with a divergence-free velocity field. In this approach, the convection term is handled as a source term in the lattice Boltzmann evolution equation; thus, the derivation term that may be induced by the convection term disappears. By using the Chapman-Enskog analysis, the anisotropic convection-diffusion

This paper introduces a novel approach for the efficient determination of confined electron states in quantum semiconductor structures through the introduction of a generalized energy eigenvalue problem formulated within the framework of boundary integral analysis. The proposed method enables the direct determination of the energy eigenvalues and normalized wavefunctions for bound quantum states. The

To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network defined on the whole problem domain. In such a partitioned neural network approach, suitable interface conditions or subdomain boundary conditions are combined to obtain

In this study, the new semilocal convergence for the family of Chebyshev Kurchatov type methods is proposed under weaker conditions. The convergence analysis demands conditions on the initial approximation, auxiliary point, and the underlying operator (Argyros et al. (2017) ). By utilizing the notion of auxiliary point in convergence conditions, the convergence domains are obtained where the existing

Computing the Casimir force and energy between objects is a classical problem of quantum theory going back to the 1940s. Several different approaches have been developed in the literature often based on different physical principles. Most notably a representation of the Casimir energy in terms of determinants of boundary layer operators makes it accessible to a numerical approach. In this paper, we

This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with

In this paper, we introduce quadratic and cubic polynomial enrichments of the classical Crouzeix–Raviart finite element, with the aim of constructing accurate approximations in such enriched elements. To achieve this goal, we respectively add three and seven weighted line integrals as enriched degrees of freedom. For each case, we present a necessary and sufficient condition under which these augmented

In this article, a fast numerical method is developed for solving the FitzHugh-Nagumo (FHN) model by combining two-grid finite element (TGFE) algorithm in space with a linearized variable-time-step (VTS) two-step backward differentiation formula (BDF2) in time. This algorithm mainly included two steps: firstly, the nonlinear coupled system on the coarse grid is solved by a nonlinear iteration; secondly

The current research, which can be employed in various engineering applications, is involved with the investigation of the heat transfer effect on the laminar and fully developed ferrohydrodynamic flow into a curved annular cylindrical duct, when a constant very strong transverse magnetic field is applied. The numerical solution of the involved constitutive partial differential equations, i.e. the

We consider the numerical computation of resonances for metallic grating structures with dispersive media and small slit holes. The underlying eigenvalue problem is nonlinear and the mathematical model is multiscale due to the existence of several length scales in problem geometry and material contrast. We discretize the partial differential equation model over the truncated domain using the finite

This study introduces an unstructured variational nodal method (UVNM-S), also recognized as the hybridized discontinuous Galerkin (HDG) method, for solving heterogeneous neutron Boltzmann transport problems. The UVNM-S solves the variational formulation of neutron Boltzmann transport equation (NBTE) by meshing the problem domain with non-overlapping nodes, i.e. the meshes. Lagrange multipliers are

In this paper, distributed strain nucleus is presented to compute the modes I/II stress intensity factors (SIFs) and electric displacement intensity factors (EDIFs) for multiple unequal cracks placed between a functionally graded piezoelectric materials (FGPMs) and a piezoelectric half-plane. Employing the Fourier transform, the governing electro-elastic equations are solved in terms of the Burgers

In this paper, a high-order direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is developed for compressible fluid flows in two-dimensional (2D) Cartesian and cylindrical coordinates. The scheme in 2D cylindrical coordinates is based on the control volume approach and it can preserve the conservation property for all the conserved variables including mass, momentum and total

In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for -elliptic problems. We first get the mixed model of -elliptic problem by introducing a new intermediate variable. Then we discuss the continuous variational problem and discrete variational problem, which based on interior penalty discontinuous Galerkin approximation. Next, we construct the

Physics-informed neural networks (PINNs) have shown remarkable achievements in solving partial differential equations (PDEs). However, their performance is limited when encountering oscillatory part in the solutions of PDEs. Therefore, this paper proposes a multi-scale deep neural network with periodic activation function to achieve high-frequency to low-frequency conversion, which can capture the

In this article, we mainly propose and analyze a parallel stabilized finite element algorithm based upon two-grid discretization for the Navier-Stokes problem. The lowest equal-order finite element pairs are considered for the finite element discretization and a stabilized term based on two local Gauss integrations is introduced to circumvent the discrete inf-sup condition. The main idea is to utilize

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear

Thin-walled structures, i.e. the wing-upper-skin panels, are prone to buckling accompanied by a significantly large out-of-plane deflection. The computational efficiency of the conventional finite element based full-order method is not satisfactory for nonlinear buckling problems of the structure. In this work, the skin panels on the upper surface of the wing butt box are selected using a sub-modeling