We propose a non-iterative monolithic projection-based method to examine the nonlinear dynamics of time-dependent chemotaxis-driven bioconvection problems. In the proposed method, all the terms are advanced using the Crank–Nicolson scheme in time along with the second-order central difference in space. Linearizations, approximate block lower–upper decompositions, and an approximate factorization technique

This paper presents a local non-singular knot method (LNKM) to accurately solve the large-scale acoustic problems in complicated geometries. The LNKM is a domain-type meshless collocation method, which relies only on scattered nodes. Firstly, a series of subdomains corresponding to every nodes can be searched based on the Euclidean distance between nodes. To each subdomain, a small linear system can

In this note, the decay estimate of solutions for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity is studied, which revises and improves a recent result obtained in [He et al. (2018), Theorem 2].

We recast a network generating partial differential equation system into a singular limit of a dissipative gradient flow model, which not only identifies the consistent physical boundary conditions but also generates networks. We then develop a set of structure-preserving numerical algorithms for the gradient flow model. Using the energy quadratization (EQ) method, we reformulate the gradient flow

The collective marking strategy with alternative refinement-indicators in adaptive mesh-refining of least-squares finite element methods (LSFEMs) has recently been shown to lead to optimal convergence rates in Carstensen (2020). The proofs utilize explicit identities for the lowest-order Raviart–Thomas and the Crouzeix–Raviart finite elements. This paper generalizes those results to arbitrary polynomial

A fast method is presented for adaptive moving mesh generation in multi-dimensions using a domain decomposition parabolic Monge–Ampère approach. The domain decomposition procedure employed here is non-iterative and involves splitting the computational domain into overlapping subdomains. An adaptive mesh on each subdomain is then computed as the image of the solution of the L2 optimal mass transfer

In this paper, we construct a finite volume element method (FVEM) on the Shishkin mesh for solving a singularly perturbed reaction–diffusion problem. The stability of the method is established in energy norm. Further, we derive the optimal error estimate in energy norm, under the decomposition of solution. Some numerical experiments are provided to confirm our theoretical results.

Melting and solidification processes are often affected by natural convection of the liquid, posing a multi-physics problem involving fluid flow, convective and diffusive heat transfer, and phase-change reactions. Enthalpy methods formulate this convection-coupled phase-change problem on a single computational domain. The governing equations can be solved accurately with a monolithic approach using

This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix–vector product involving the mass matrix itself. We prove that the condition

Building on recent advances in the analysis and design of algebraic flux correction (AFC) schemes, new differentiable limiter functions are constructed for efficient nonlinear solution strategies. The proposed scaling parameters are used to limit artificial diffusion operators incorporated into the residual of a high order target scheme to produce accurate and bound-preserving finite element approximations

The nonlinear time-fractional stochastic fourth-order reaction–diffusion equation perturbed by noises is considered by the mixed finite element method in this paper. Based on the mixed finite element in spatial direction and the generalized BDF2−θ in temporal discretization, the semi- and fully-discrete schemes are obtained. Further, the error estimates for the semi- and fully-discretizations and the

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson

We present an efficient isogeometric analysis (IGA) formulation for nonlinear vibration problems of viscoelastic plates. The formulation is based on higher-order shear deformation theory (HSDT), which is guaranteed by p-order nonuniform rational B-splines (NURBS) resulting in Cp−1 continuity in the domain. The governing equations of motion are derived from Hamilton’s principle. The generalized multi-axial

We study a mixed discontinuous Galerkin (DG) method for solving the second-order wave equation. The stress variable p and the displacement variable u are discretized by the mixed DG element pair Pk+1–Pk (k≥0). Under appropriate regularity assumptions on the solution pair, we derive optimal error estimates for the spatially semi-discrete scheme. Specifically, we prove the optimal convergence orders

In this paper, we derive an L1 Legendre–Galerkin spectral method with fast algorithm based on an efficient sum-of-exponentials (SOE) approximation for the kernel t−1−α to solve the two-dimensional nonlinear coupled time fractional Schrödinger equations. The numerical method is stable without the Courant–Friedrichs–Lewy (CFL) conditions based on the error splitting argument technique and the discrete

In this paper, several active set methods based on classical problems arising in Contact Mechanics are analyzed, namely unilateral/bilateral contact associated with Tresca’s/Coulomb’s law of friction in small and large deformation. The purpose of this work is to extend an Inexact Primal–Dual Active Set (IPDAS) method already used in Hueber et al. (2008) to the formalism of dynamics and hyper-elasticity

A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1(Ω)-conforming Taylor–Hood finite element combination, consisting of polynomial degrees k+1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori

A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable v=Δu, the fourth-order problem is reduced into a couple of second-order system

In this paper, a system of partial differential equations, that can be used to describe the drug release from a biodegradable viscoelastic polymeric platform, is studied from an analytical and numerical point of view. The system is defined in a moving boundary domain and its stability is analysed. From numerical point of view, a numerical method is proposed and its convergence properties are established

We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS-FE) method (Calo et al., 2020) for scalar-valued convection–diffusion problems. The AVS-FE method is a Petrov–Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C0 or Raviart–Thomas functions. We employ the concept of

In this paper, an efficient splitting domain decomposition method scheme for solving time-dependent convection–diffusion reaction equations is analyzed. A three-step method along each direction is used to solve the solution over each block-divided sub-domain at every time interval. The new solutions are firstly solved by the quadratic interpolation. Then, the intermediate fluxes on the interfaces of

In this paper, a kind of finite-difference lattice Boltzmann method with the second-order accuracy of time and space (T2S2-FDLBM) is proposed. In this method, a two-stage time-accurate discretization approach is applied to construct time marching scheme, and the spatial gradient operator is discretized by a mixed difference scheme to maintain a second-order accuracy in space. It is shown that the previous

Solidification heat transfer process of billet is described by nonlinear partial differential equation (PDE). Due to the poor productive environment, the boundary condition of this nonlinear PDE is difficult to be fixed. Therefore, the identification of boundary condition of two-dimensional nonlinear PDE is considered. This paper transforms the identification of boundary condition into a PDE optimization

The stationary Darcy–Brinkman equations in the double-diffusive convection, which model the heat and mass transfer phenomena, are considered in this paper. Based on a suitable contractive operator, the existence and uniqueness of the problem are firstly proved by using the fixed point theorem. The regularities of the weak solution are also derived. Then, the Newton iterative method is studied for solving

Solutions of 3D thermal analysis in inhomogeneous media with variable thermal source are divided into homogeneous and particular solutions by a novel boundary-type element-free method, namely virtual boundary element-free Galerkin method (VBEFGM). The homogeneous solution can be obtained by the virtual boundary element-free method (VBEFM). The virtual source function of the homogeneous solution and

This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery

In this paper, we shall present the alternating direction implicit (ADI) Galerkin finite element method (FEM) for solving the distributed-order time-fractional mobile–immobile equation in two dimensions. In the time direction, the backward Euler method is used to deal with the temporal first-order derivative, and the weighted and shifted Grünwald formula is employed to discretize the distributed-order

In magnetic resonance imaging of the human brain, the diffusion process of tissue water is considered in the complex tissue environment of cells, membranes and connective tissue. Models based on fractional order Bloch–Torrey equations are known to provide insights into tissue structures and the microenvironment. In this paper, we consider new two-dimensional multi-term time and space fractional Bloch–Torrey

In this paper, we propose two finite difference schemes to solve a class of a fourth-order strongly damped nonlinear wave equation in two dimensions. We prove some a priori bounds and establish the second order convergence in L∞−norm for the difference solutions. We also discuss the stability and the unique solvability of the two proposed difference schemes. Finally, we provide numerical examples to

We consider an adaptive finite element method for solving parabolic interface problems with nonzero flux jumps in a two-dimensional convex polygonal domain. We use continuous, piecewise linear functions for the approximation of the spatial variable whereas the backward Euler method is employed for the time discretization. The reliability bound of the estimator is derived in terms of the error indicators

We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier–Stokes equations. Our method employs a technique developed by Guermond and Minev (2011), which consists of singular perturbation plus a splitting scheme. While the time-integration schemes are implicit, we use finite elements to discretize the spatial counterparts. At each time-step

This work concerns the weights of the radial basis function generated finite difference (RBF-FD) formulas for estimation of the first and second derivatives of an unknown function applying the generalized multiquadric function (GMQ). Several discussions about their error equations on structured and unstructured grids of points are worked out. Next, the formulas are applied on a new non-uniform mesh

Stability and error estimates for a projection based variational multiscale finite element scheme for Oseen problem in a time-dependent domain are derived in this paper. The use of Geometric Conservation Law (GCL) provides an unconditional stable scheme, whereas a restriction on the time-step needs to be imposed to obtain stability estimates independent of the mesh velocity when GCL is violated. Further

Phase-field methods have the potential to simulate large scale evolution of networks of fractures in porous media without the need to explicitly track interfaces. Practical field simulations require however that robust and efficient decoupling techniques can be applied for solving these complex systems. In this work, we focus on the mechanics-step that involves the coupling of elasticity and the phase-field

The present work is to develop the FMBEM with quadratic elements to the analysis of 3-D linear elastic structures containing subdomains. To the numerical integration, an adaptive method with subdivision of elements is applied. The BEM formulation for perfectly bonded subdomains, in which interface traction forces are eliminated from the system of equations is implemented. The method is applied to problems

We present a high-order entropy stable discontinuous Galerkin (ESDG) method for the two dimensional shallow water equations (SWE) on curved triangular meshes. The presented scheme preserves a semi-discrete entropy inequality and remains well-balanced for continuous bathymetry profiles. We provide numerical experiments which confirm the high-order accuracy and theoretical properties of the scheme, and

In this paper, finite point method (FPM) and meshless weighted least squares (MWLS) method are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The FPM scheme uses shape function constructed by moving least square (MLS) approximation to discretize the equations, while the MWLS scheme employs both MLS approximation and penalty terms to solve the same problem

We describe a fully discrete mixed finite element method for the linearized rotating shallow water model, possibly with damping. While Crank–Nicolson time-stepping conserves energy in the absence of drag or forcing terms and is not subject to a CFL-like stability condition, it requires the inversion of a linear system at each step. We develop weighted-norm preconditioners for this algebraic system

We present an algorithm for numerical simulation of electric current in rock samples and numerical upscaling of heterogeneous samples’ electrical resistivity (conductivity). The solver is oriented on solving strongly heterogeneous problems so that partially saturated multi-mineral core samples can be treated. The algorithm is based on the Krylov-type solver, where the preconditioner is the inverse

In this work, we prove the convergence of a discretized anisotropic degenerate parabolic system modeling chemotaxis. The used method belongs to a positive nonlinear Discrete Duality Finite Volume (DDFV) family. Its aim is to respect the physical bounds on the computed solution independently of the used mesh and the involved anisotropy. To achieve that, two main terms referred to as convection and diffusion

We study the Darcy boundary value problem with lognormal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor

In this paper, we propose a linear, unconditional energy stable time discretization scheme for Cahn–Hilliard–Navier–Stokes model, which is a phase-field model for two-phase incompressible flow. Based on a Lagrange multiplier approach, our proposed scheme is linearized by using implicit–explicit treatments, and the computation of the velocity field u, the pressure p, the phase function ϕ are decoupled

We consider preconditioned iterative methods for numerical solution of the convective FitzHugh–Nagumo equations. By eliminating the symmetric indefinite matrix derived from the single nonlinear term of the convective FitzHugh–Nagumo equations, a preconditioner that is robust with respect to the involved model parameters is proposed to solve the arising discretized linear system. Spectral properties

A linear triangular finite element method-implicit difference scheme (FEM-IDS) for the solution of second order strongly damped wave equation (SDWE) with memory on domain with interfaces is proposed. Sufficient conditions that guarantee the existence of a unique solution are given. The finite element discretization is such that the arbitrary (but smooth) interface is first approximated by a polygon

The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model

Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix which converts primal representations to dual representations

The work is concerned with two problems: (a) analysis of a discontinuous Petrov–Galerkin (DPG) method set up in fractional energy spaces, (b) use of the results to investigate a non-conforming version of the DPG method for general polyhedral meshes. We use the ultraweak variational formulation for the model Laplace equation. The theoretical estimates are supported with 3D numerical experiments.

A least-squares formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement (LS-MDG-ICE) is presented. This method combines MDG-ICE, which uses a weak formulation that separately enforces a conservation law and the corresponding interface condition and treats the discrete geometry as a variable, with the Discontinuous Petrov–Galerkin (DPG) methodology

We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in

In this work, a dynamic-Immersed–Boundary method combined with a BGK-Lattice–Boltzmann technique is developed and critically discussed. The fluid evolution is obtained on a three-dimensional lattice with 19 reticular velocities (D3Q19 computational molecule) while the immersed body surface is modeled as a collection of Lagrangian points responding to an elastic potential and a bending resistance. A

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale

We develop a new computational model to describe hydro-mechanical coupling in fractured rocks composed of a linear poroelastic Biot medium and nonlinear elastic joints with constitutive response governed by the Barton–Bandis (BB) law. The model aims at capturing increase in stiffness induced by fracture closure during fluid withdrawal. The nonlinear hydro-mechanical formulation is constructed within

This work is concerned with the construction and the hp non-conforming a priori error analysis of a Discontinuous Galerkin DG numerical scheme applied to the hypersingular integral equation related to the Helmholtz problem in 3D. The main results of this article are an error bound in a norm suited to the problem and in the L2-norm. Those bounds are quasi-optimal for the h-convergence and the p-convergence

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting

In this paper, we propose a reduced-order method (ROM) based on the Monte Carlo finite difference method (FDM) or Monte Carlo finite element method (FEM) for the stochastic Allen–Cahn (SAC) equation with multiplicative noise. The reduction method is a combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). In the theoretical part, the error bounds of

We present a three-dimensional (3D) partitioned aeroelastic formulation for a flexible multibody system interacting with incompressible turbulent fluid flow. While the incompressible Navier–Stokes system is discretized using a stabilized Petrov–Galerkin procedure, the multibody structural system consists of a generic interaction of multiple components such as rigid body, beams and flexible thin shells

This paper presents a novel GPU-parallelized meshless method for solving Reynolds-averaged Navier–Stokes equations with the Spalart–Allmaras turbulence model. Least-square curve fit is utilized to discretize the spatial derivatives of the equations, and a Roe-type upwind scheme is used for computing the flux terms. The compute unified device architecture (CUDA) Fortran programming model is employed

In this paper we present a seven-velocity three-dimensional (D3N7) vectorial lattice Boltzmann method (LBM) including various types of approximations to the pressure and propose a family of two-parameterized second-order boundary schemes with accuracy independent of the boundary location. In order to show the numerical stability of the D3N7 model, we construct a symmetrizer to handle the nonlinear

A novel procedure is given for choosing smoothest stencil to construct less oscillatory ENO schemes. The procedure is further used to define the smoothness parameter in the non-linear weights of new WENO schemes. The main significant features of these new ENO and WENO schemes are that they are less oscillatory and achieve their relevant order of accuracy in the presence of critical points in the exact

In this study, a numerical investigation on nanofluid flow and heat transfer is presented in a cavity under the effects of cross partial magnetic fields and a partial heater. The governing stream function–vorticity equations are solved by thin plate spline radial basis functions (Rbfs) in space derivatives and backward Euler method in time derivatives. The vorticity transport equation involves either