This study explores the use of r-adaptive mesh refinement strategies for elliptic partial differential equations (PDEs) posed on non-convex domains. We introduce an r-adaptive strategy based on a simplified optimal transport method to create a graded mesh, distributing the interpolation error evenly, considering the solution's local asymptotic behaviour. The grading ensures good mesh compression and

This work aims to construct an efficient and highly accurate numerical method to address the time singularity at t=0 involved in a class of time-fractional parabolic integro-partial differential equations in one and two dimensions. The L2-1σ scheme is used to discretize the time-fractional operator, whereas a modified version of the composite trapezoidal approximation is employed to discretize the

Via introducing the Robin flux jump into the Galerkin scheme, this paper develops a new anisotropic trilinear immersed finite element (IFE) method for solving the magnetized plasma diffusion problem with plasma sheath interface condition under Cartesian meshes. The three-dimensional (3D) diffusion process of magnetized plasma is anisotropic and highly sensitive to magnetic fields, making it difficult

In this work, we deal with an inverse source problem for a general transport equation. First, we discuss the solvability of the problem. Next, in order to solve the problem, we propose a new hybrid numerical algorithm which is based on the finite difference method, Newton-Cotes formula, Lagrange polynomial approximation and composite trapezoidal rule. The proposed method is tested on several examples

We present the numerical analysis of a finite element method (FEM) for one-dimensional Dirichlet problems involving the logarithmic Laplacian (the pseudo-differential operator that appears as a first-order expansion of the fractional Laplacian as the exponent s→0+). Our analysis exhibits new phenomena in this setting; in particular, using recently obtained regularity results, we prove rigorous error

In this paper, a Nitsche's extended nonconforming virtual element method is presented to discretize biharmonic PDEs involving interfaces with a more general interface condition. By introducing some special terms on cut edges and uncut edges of interface elements, we prove the well-posedness and optimal convergence, which are independent of the location of the interface relative to the mesh and the

A novel adaptive multi-patch isogeometric approach based on truncated hierarchical NURBS (TH-NURBS) is proposed for modeling three-dimensional elasticity. The TH-NURBS are rational extension of truncated hierarchical B-splines (THB-splines) and the salient feature of the TH-NURBS is that it can exactly model complex-shaped geometries. Owing to the properties of local refinement, partition-of-unity

The a posteriori error estimator using the least-squares functional can be used for adaptive mesh refinement and error control even if the numerical approximations are not obtained from the corresponding least-squares method. This suggests the development of a versatile non-intrusive a posteriori error estimator. In this paper, we present a systematic approach for applying the least-squares functional

The paper compares standard iterative methods for solving the generalized Stokes problem arising from the time and space approximation of the time-dependent incompressible Navier-Stokes equations. Various preconditioning techniques are considered: (1) pressure Schur complement; (2) fully coupled system using an exact factorization as a basis for the preconditioner; (3) fully coupled system using norm

This work addresses the inverse problem of recovering the time-varying fractional order α(t) in a time-fractional diffusion equation, motivated by applications in subsurface flows and shale gas extraction. The fractional order α(t) plays a crucial role in modeling anomalous diffusion processes, such as those observed in complex geological formations. Prior to developing the reconstruction method, the

Based on the block Jacobi splitting, a kind of greedy partial block Jacobi (GPBJ) iteration method is constructed by greedily selecting the blocks with relatively large residuals and performing the block Jacobi iteration on the selected blocks. Theoretical analysis demonstrates that the GPBJ iteration is unconditionally convergent if the coefficient matrix of the linear system is H-matrix. Then combining

The accurate thermal analysis of composite structures remains a challenging issue due to complicated multiscale configurations and nonlinear temperature-dependent behaviors. This work offers a novel higher-order three-scale asymptotic (HOTSA) model and corresponding numerical algorithm for accurately and efficiently simulating transient nonlinear thermal conduction problems of heterogeneous structures

For high-order accurate schemes such as discontinuous Galerkin (DG) methods solving Fokker–Planck equations, it is desired to efficiently enforce positivity without losing conservation and high-order accuracy, especially for implicit time discretizations. We consider an optimization-based positivity-preserving limiter for enforcing positivity of cell averages of DG solutions in a semi-implicit time

A novel arbitrary high-order energy-stable fully discrete schemes are proposed for the nonlinear Benjamin-Bona-Mahony-Burgers equation based on linearized Crank-Nicolson scheme in time and the virtual element discretization in space. Two skew-symmetric discrete forms are introduced to preserve energy dissipation of the numerical scheme. Furthermore, by utilizing the L2 projection to approximate the

In this paper, we present the coupled nonlinear Schrödinger equations to describe the conformational dynamics of protein secondary structure. We first construct a structure-preserving discrete scheme that ensures both mass conservation and energy stability. The proposed scheme is employed by combining the discontinuous Galerkin (DG) method for spatial discretization, Crank-Nicolson (C-N) approximation

The smoothed finite element model exhibits a "softening effect," resulting in reduced stiffness compared to the standard finite element model. This study employs the smoothed finite element methods (S-FEMs) with automatically generated tetrahedral meshes to perform modal analysis of biological structures subjected to arbitrary dynamic forces. Various S-FEM models are developed, including Edge-based

The dispersion error, also known as the pollution effect, is one of the main difficulties in numerical solutions to the wave propagation problem at high wavenumbers. The pollution effect, especially in mesh-based methods, can potentially be controlled by using either finer meshes or higher-order discretizations. Using finer meshes often leads to large systems that are computationally expensive to solve

This work addresses the development of a physics-informed neural network (PINN) with a loss term derived from a discretized time-dependent full-order and reduced-order system. In this work, first, the governing equations are discretized using a finite difference scheme (whereas any other discretization technique can be adopted), then projected on a reduced or latent space using the Proper Orthogonal

Aerosols, which are generated by the rupture of the liquid plug in the pulmonary respiratory tract, are important carriers of the viruses of infectious respiratory diseases, such as flu, tuberculosis, COVID-19, and Measles. In this study, we investigate liquid plug rupture and aerosol generation in the low respiratory tract with the alveolar structures by the chemical-potential multiphase lattice Boltzmann

We present a new discretization scheme to solve the stationary drift-diffusion equations based on the hybrid mixed finite element method. A convenient change of variables is adopted and the partial differential equations of the system are decoupled and linearized through Gummel's map. This gives rise to three equations that need to be solved in a staggered fashion: one of reaction-diffusion type (Poisson)

In this article, an energy dissipative Crank-Nicolson (C-N) type fully discrete nonconforming finite element method (FEM) is developed for the damped wave equation with cubic nonlinearity, and its unconditional superconvergence behavior is rigorously analyzed for the nonconforming EQ1rot element. By introducing an auxiliary variable p=ut, the problem is converted to a proper parabolic system, a new

In this article, we aim to develop a high order direct discontinuous Galerkin (DDG) method solving elliptic interface problem on arbitrary polygon fitted meshes. Elliptic interface problem with the homogeneous or non-homogeneous interface conditions can be solved in the uniform discrete DDG formulation. Numerical analysis results show that high order DDG method for polygonal elliptic interface problem

In this paper, we propose a novel numerical approach to model the complex blood flow in microvessels using a two-layered fluid representation. The model considers blood flow as two layers of homogeneous, immiscible fluid with different viscosities: a core layer, rich in erythrocytes (red blood cells, RBCs), occupying the central region of the vessel, and a peripheral cell-free plasma layer (CFL) near

Shell structures, characterized by their thin walls, are prone to significant displacements and deformations under loading, leading to geometric nonlinearity while local deformations or strains remain small. The corotational (CR) method, where the total motion is separated into rigid body motion and elastic displacement, effectively simplifies such complex large-rotation problems into local small strain-small

This study investigates the free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates using a novel approach. This approach employs Chebyshev polynomials to represent both new shear deformation theory and moving Kriging meshfree shape functions. The proposed theory, termed the third-order Chebyshev shear deformation theory (TCSDT), automatically fulfills

This paper describes a numerical scheme for solving a reaction-diffusion system, specifically the Gray-Scott model. The scheme is a two-step process, combining the Crank-Nicolson method in the first step with the second-order backward differentiation formula in the second step. This combination ensures unconditional stability in both L2 and H1-norms and allows for optimal error estimates. The scheme's

This paper introduces a novel technique for addressing singular kernels in the variational boundary element formulation. The study presents an expansion centered on two explicit singular points, enabling the isolation of all relevant singular terms and extending the analytical capabilities to curved elements, thereby broadening the applicability of the formulation. The proposed method comprehensively

In this paper, we study the two-step backward differentiation formula (BDF2) finite element method for the two-dimensional time-dependent linear Schrödinger equation. Firstly, we obtain the BDF2 fully discrete finite element scheme of the Schrödinger equation, and analyze unconditional optimal error estimates by dividing the error analysis into temporal error and spatial error analysis, respectively

Numerical methods of approximate solution of the Cauchy problem for coupled systems of evolution equations are considered. Separating simpler subproblems for individual components of the solution achieves simplification of the problem at a new level in time. The decoupling method, a significant approach to simplifying the problem, is based on the decomposition of the problem's operator matrix. The

In this paper, we propose a linear second-order numerical method for solving the Allen-Cahn equation with general mobility. The fully-discrete scheme is achieved by using the Crank-Nicolson formula for temporal integration and the central difference method for spatial approximation, together with two additional stabilization terms. Under mild constraints on the two stabilizing parameters, the proposed

We develop a recursive integration formula for a class of rational polynomials in 2D. Based on this, we present implementations of finite elements that have rational basis functions. Specifically, we provide simple MATLAB implementations of the singular Zienkiewicz and the lowest-order Guzmán–Neilan finite element in 2D.

We present a meshless finite difference method for multivariate scalar conservation laws that generates positive schemes satisfying a local maximum principle on irregular nodes and relies on artificial viscosity for shock capturing. Coupling two different numerical differentiation formulas and the adaptive selection of the sets of influence allows to meet a local CFL condition without any a priori

In this article, plain convergence of an adaptive finite element method is shown for a non-local problem of Kirchhoff type under the same assumptions on the data as in the paper of T. Gudi (2012) [21]. Then by imposing some additional assumptions on the data, convergence and quasi-optimality of an AFEM is proved for the Kirchhoff type problem. The theoretical results are illustrated by some numerical

An adaptive moving mesh or r-refinement method for the numerical approximation of pitting corrosion in heterogeneous materials is designed and applied to the problem of pitting corrosion in metals. The pitting corrosion is described by Laplace's equation with a moving boundary where the moving boundary problem is coupled with the partial differential equations describing the mesh movement. We show

An improved boundary knot method (IBKM) is proposed to enhance the performance of BKM in solving homogeneous high-order Helmholtz-type partial differential equations. Compared with the classical BKM where the sources are always placed on the physical boundary as collocation points, the new sources named fictitious points are now placed on multi-layer extended pseudo boundaries. This modification leads

For the fourth order time-dependent singularly perturbed Bi-wave equation modeling d-wave superconductors, the implicit Backward Euler (BE) and Crank-Nicolson (CN) schemes of Galerkin finite element method (FEM) are studied by Bonner-Fox-Shmite element. Then the quasi-uniform and unconditional superconvergent error estimates of orders O(h3+τ) and O(h3+τ2) (h, the spatial parameter, and τ, the time

We develop a virtual element method to solve a convective Brinkman-Forchheimer problem coupled with a heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions and appropriate assumptions on the data, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under

This paper primarily focuses on developing a high-order-in-time spectral Galerkin approximation method for nonlinear stochastic Fisher's type equations driven by multiplicative noise. For this reason, we first design an improved discretization scheme in time based on the Milstein method, and then propose a spectral Galerkin approximation method in space. We analyze the H1 stability and L2 stability

This article aims at studying new finite difference methods for one-dimensional and two-dimensional nonlinear Riesz space variable-order (VO) fractional diffusion equations. In the presented model, fractional derivatives are defined in the Riemann-Liouville type. Based on 4-point weighted-shifted-Grünwald-difference (4WSGD) operators for Riemann-Liouville constant-order fractional derivatives, which

In this paper, we consider higher order multipoint flux mixed finite element methods for parabolic problems. The methods are based on enhanced Raviart-Thomas spaces with bubbles. The tensor-product Gauss-Lobatto quadrature rule is employed, which enables local velocity elimination and results in a symmetric, positive definite cell-based system for pressures. We construct two fully discrete schemes

In this paper, we present and analyze a second-order exponential time differencing Runge–Kutta (ETDRK2) scheme for the FitzHugh-Nagumo equation. Utilizing a second-order finite-difference space discretization, we derive the fully discrete numerical scheme by incorporating both the stabilization technique and the ETDRK2 scheme for temporal approximation. The smallest invariant set of the FitzHugh-Nagumo

In the analysis of transient seepage flow with free surfaces, not only the free surfaces but also the boundary conditions vary with time, introducing significant challenges to those traditional mesh-based numerical methods. Although the numerical manifold method (NMM) has shown great advantages in tracking time-independent free surface seepage flow due to its dual cover systems – the mathematical cover

Chew, Goldberger & Low (CGL) equations describe one of the simplest plasma flow models that allow anisotropic pressure, i.e., pressure is modeled using a symmetric tensor described by two scalar pressure components, one parallel to the magnetic field, another perpendicular to the magnetic field. The system of equations is a non-conservative hyperbolic system. In this work, we analyze the eigensystem

This paper investigates the uniform convergence of arbitrary order finite element methods on Vulanović-Bakhvalov mesh. We carefully design a new interpolation based on exponential layer structure, which not only overcomes the difficulties caused by the mesh step width, but also ensures the Dirichlet boundary condition. We successfully demonstrate the uniform convergence of the optimal order in the

The objective of this paper is to present efficient numerical algorithms to resolve the two-dimensional fractional Oldroyd-B model. Firstly, two compact alternating direction implicit (ADI) methods are constructed with convergence orders O(τmin⁡{3−γ,2−β,1+γ−2β}+hx4+hy4) and O(τmin⁡{3−γ,2−β}+hx4+hy4), where γ and β are orders of two Caputo fractional derivatives, τ, hx and hy are the time and space

This article introduces an efficient numerical method for solving the Black-Scholes partial differential equation (PDE) that governs European options. The methodology employs the backward Euler scheme to discretize the time derivative and incorporates the non-symmetric interior penalty Galerkin method for handling the spatial derivatives. The study aims to determine optimal order error estimates in

Advection-diffusion-reaction (ADR) models describe transport mechanisms in fluid or solid media. They are often formulated as partial differential equations that are spatially discretized into systems of ordinary differential equations (ODEs) in time for numerical resolution. This paper investigates the performance of variable stepsize, semi-implicit, backward differentiation formula (VSSBDF) methods

Anisotropic fluids, such as nematic liquid crystals, can form non-spherical equilibrium shapes known as tactoids. Predicting the shape of these structures as a function of material parameters is challenging and paradigmatic of a broader class of problems that combine shape and order. Here, we consider a discrete shape optimization approach with finite elements to find the configuration of two-dimensional

In this paper, we explore efficient methods for discretized linear systems that arise from eddy current optimal control problems utilizing an all-at-once approach. We propose a novel low-rank matrix equation method based on a special splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK) algorithm. First, we reformulate the resulting discretized linear system into a matrix equation

In this work, we apply the local Petrov-Galerkin method based on radial basis functions to solving the two dimensional linear hyperbolic equations with variable coefficients subject to given appropriate initial and boundary conditions. Due to the presence of variable coefficients of the differential operator, special treatment is carried out in order to apply Green's theorem and derive the variational

Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which remain fixed throughout the computation, and uses a linear least squares method for training the parameters of the output layer of the neural network. It is known

The purpose of this article is to explore the superconvergence behavior of the first-order backward-Euler (BE) implicit/explicit fully discrete schemes for the unsteady incompressible MHD equations with low-order mixed finite element method (MFEM) by utilizing the scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) methods. Through dealing with linear terms in implicit format and nonlinear

The Hamilton-Jacobi(HJ) equation represents a class of highly nonlinear partial differential equations. Classical numerical techniques, such as finite element methods, face significant challenges when addressing the numerical solutions of such nonlinear HJ equations. However, recent advances in neural network-based approaches, particularly Physics-Informed Neural Networks (PINNs) and neural operator

In this paper, we propose an innovative isogeometric low-rank solver for the linear elasticity model problem, specifically designed to allow multipatch domains. Our approach splits the domain into subdomains, each formed by the union of neighboring patches. Within each subdomain, we employ Tucker low-rank matrices and vectors to approximate the system matrices and right-hand side vectors, respectively

Fully discrete virtual element methods with second-order accuracy in temporal direction require to choose smaller time steps in order to maintain the higher accuracy provided by the spatial direction. To overcome this restriction higher order time stepping methods are needed. In this work the general Newmark scheme for temporal discretization is considered along with the virtual element discretization

Using the T-coercivity theory as advocated in Chesnel and Ciarlet (2013) [25], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when

A novel two-grid symmetric mixed finite element analysis is considered for semi-linear second order hyperbolic problem. To overcome the saddle-point problem resulted by the traditional mixed element methods, a new symmetric and positive definite mixed procedure is first introduced to solve semi-linear hyperbolic problem. Then the a priori error estimates both in L2 and Lp-norm senses are derived. Meanwhile

The paper presents the average radial particle method (ARPM), a new mesh-free technique for solving partial differential equations (PDEs). Here, we use the ARPM to solve 3D transient heat transfer problems. ARPM numerically approximates spatial derivatives by discretizing the domain by particles such that each particle is only affected by its direct neighbors. One feature that makes ARPM different