In this paper, we consider the final boundary value problem for non-local Kirchhoff’s model of parabolic type with discrete random noise. We first discuss the instability of solutions. Then we present the regularized solution by the trigonometric method in non-parametric regression associated with the truncated expansion method. In addition, under a prior assumption on the exact solution, the convergence

Rough polyharmonic splines (RPS) is a variational method which has recently been developed for linear divergence-form operators with arbitrary rough coefficients. RPS method does not rely on concepts of ergodicity or scale-separation, but on compactness properties of the solution space. In this paper, we extend RPS approach method for the optimal control problem governed by parabolic systems with rough

Finite element analysis (FEA), introduced more than half a century ago, requires a (qualified) analyst to generate the necessary input data, verify the output and post process the analysis results for a meaningful conclusion. The required expertise and labor efforts have precluded the use of FEA in daily medical practice by orthopedic surgeons for example. Patient-specific analyses of the mechanical

We present a parallel data structure for the discretization of partial differential equations which is based on distributed point objects and which enables the flexible, transparent, and efficient realization of conforming, nonconforming, and mixed finite elements. This concept is realized for elliptic, parabolic and hyperbolic model problems, and sample applications are provided by a tutorial complementing

A reaction–diffusion system is formulated to describe three interacting species within the Hastings–Powell (HP) food chain structure with chemotaxis produced by three chemicals. We construct a finite volume (FV) scheme for this system, and in combination with the non-negativity and a priori estimates for the discrete solution, the existence of a discrete solution of the FV scheme is proven. It is shown

Topology optimization is widely applied to various design problems in both structure and fluid dynamics engineering. Specifically, the development of energy dissipation devices with vibration control remains a key consideration. The aim of this study is to improve devices that maximize the absorption or dissipation of the vibration of an oscillating object and propose an approach in which the level

Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported, single-walled carbon nanotubes (SWCNTs) non-embedded or embedded in an elastic medium using the multiple time scale method. Eringen’s nonlocal constitutive relations are adopted to account for the small length scale

T-IFISS is a finite element software package for studying finite element solution algorithms for deterministic and parametric elliptic partial differential equations. The emphasis is on self-adaptive algorithms with rigorous error control using a variety of a posteriori error estimation techniques. The open-source MATLAB framework provides a computational laboratory for experimentation and exploration

In this paper, we construct and analyze a family of quadratic finite volume element schemes over triangular meshes for elliptic equations. This family of schemes cover some existing quadratic schemes. For these schemes, by element analysis, we find that each element matrix can be split as two parts : the first part is the element stiffness matrix of the standard quadratic finite element method, while

Assume that u(x) satisfies the problem Lu(x)≡−∂∂xi(aij∂u∂xj)=f(x),∀x∈Ω,u(x)=0,∀x∈∂Ω. In this article, using interpolation postprocessing technique, we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u(x) using piecewise polynomials of degrees bi-k(k≥3) over a rectangular partition. Assume that k≥3 is odd and x0 is an interior

The aim of this paper is to propose and analyze a numerical method to solve transient eddy current problems formulated in terms of the magnetic field intensity. Space discretization is based on Nédélec edge elements, while a backward Euler scheme is used for time discretization; the curl-free constraint in the dielectric domain is imposed by means of a penalty strategy. Convergence of the penalized

Capillary pressure is a key parameter in reservoir numerical simulation. It is generally considered that the capillary force is an only function of water saturation in the current traditional reservoir numerical simulation methods. However, large amount of indoor experiments have shown that capillary force is not only dependent on the water saturation, but also the velocity of fluid flow. In this work

A block triple-relaxation-time (B-TriRT) lattice Boltzmann model for general nonlinear anisotropic convection–diffusion equations (NACDEs) is proposed, and the Chapman–Enskog analysis shows that the present B-TriRT model can recover the NACDEs correctly. There are some striking features of the present B-TriRT model: firstly, the relaxation matrix of B-TriRT model is partitioned into three relaxation

The paper discusses the efficiency of the classical BiCGStab method and several of its modifications for solving systems with multiple right-hand side vectors. These iterative methods are widely used for solving systems with large sparse matrices. The paper presents execution time analytical model for the time to solve the systems. The BiCGStab method and several modifications including the Reordered

In this paper, we have constructed a scheme combining generalized Polynomial Chaos (gPC) representation and B-spline wavelets. To begin with, semi-orthogonal compactly supported B-spline wavelets are constructed for the bounded interval [0,1] which are used for PC expansion of possible stochastic processes. To compute the deterministic coefficients of expansion, we have applied Galerkin projection

In this paper, the idea of direct discretization via radial basis functions (RBFs) is applied on a local Petrov–Galerkin test space of a partial differential equation (PDE). This results to a weak-based RBF-generated finite difference (RBF–FD) scheme that possesses some useful properties. The error and stability issues are considered. When the PDE solution or the basis function has low smoothness,

Open cell metal foams are interesting lightweight materials, which are used in a variety of applications. Effected by the complex microstructure of open cell metal foams the macroscopic material behaviour cannot be described with a von Mises yield criterion. In comparison to bulk metals, open cell metal foams collapse under hydrostatic compression and tension. Modelling of this behaviour is realised

We develop a novel Hybrid High-Order method for the incompressible Navier–Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective contributions in the momentum equation formulated in terms of a globally divergence-free velocity reconstruction. Two key properties are mimicked at the discrete level, namely the

The purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry. These approximations are based on the application of enriched versions of classic Poisson-compatible spaces, for stress and displacement variables, and/or on enriched Stokes-compatible space configurations, for the choice of rotation spaces used to weakly

A PDE model based on an image specific auto generated multi-well potential function and a 4th order anisotropic diffusion with good performance with respect to loss due to smoothing effects has been proposed for grayscale image inpainting. An unconditionally stable numerical scheme using the notion of convexity splitting in time and Fourier spectral method in space has been derived. The stable scheme

In this paper, we consider an underlying general stochastic volatility jump-diffusion model. Option pricing under this general model with transaction costs will lead to handling with nonlinear partial integro-differential equations (here after PIDE). In this case, option replication in a discrete-time framework with transaction costs and the non-uniqueness option pricing in such incomplete market will

A high-order three-scale approach developed in this work to analyze thermo-mechanical properties of porous materials with interior surface radiation is systematically studied. The microstructures of the porous structures are described by periodical layout of local cells on the microscopic domain and mesoscopic domain, and surface radiation effect at microscale and mesoscale is also investigated. At

A fully discrete two-grid modified method of characteristics (MMOC) scheme is proposed for nonlinear variable-order time-fractional advection–diffusion equations in two space dimensions. The MMOC is used to handle the advection-dominated transport and the two-grid method is designed for efficiently solving the resulting nonlinear system. Optimal L2 error estimates are derived for both the MMOC scheme

A time-fractional initial–boundary value problem is considered, where the spatial domain has dimension d∈{1,2,3}, the spatial differential operator is a standard elliptic operator, and the time derivative is a Caputo derivative of order α∈(0,1). To discretise in space we use a standard piecewise-polynomial finite element method, while for the temporal discretisation the GMMP scheme (a variant of the

We present the usage of an adjoint Lattice Boltzmann Method (LBM) for open-loop control of two-dimensional flapping wing motion. Movement of the wing is parametrised with periodic B-Splines, while the wing interacts with the surrounding flow via an Immersed Boundary (IB) method. Multi-objective optimisation is performed using a gradient optimisation algorithm, for which sensitivities are calculated

Cryosurgery is a surgical technique to ablate abnormal tumor tissues with the help of extreme cold using cryogens such as liquid nitrogen, solid carbon dioxide, argon etc. This therapy may be used as a primary treatment for early prostate cancers effectively. During the process of freezing in cryosurgery, a phase transition occurs between the frozen tumor tissues and unfrozen surrounding tissues. The

In this paper, the original finite volume lattice Boltzmann method (FVLBM) on an unstructured grid (Part I of these twin papers) is extended to simulate turbulent flows. To model the turbulent effect, the k−ω SST turbulence model is incorporated into the present FVLBM framework and is also solved by the finite volume method. Based on the eddy viscosity hypothesis, the eddy viscosity is computed from

We propose a radial basis function collocation method (RBF method) to solve fractional Laplacian visco-acoustic wave equation for the Earth media having heterogeneous velocity model and complex geometry. Unlike the fractional Laplacian wave equation proposed in Zhu and Harris (2014), the wave equation we consider has a different definition for the fractional Laplacian. Specifically, spectral and Riesz

We construct a new scheme whose primary unknowns include both cell-centered unknowns and edge unknowns on discontinuous line to solve diffusion equations with discontinuous coefficient. First, two linear fluxes are given on both sides of the cell edge, respectively. In order to deal with the defect of the existing scheme preserving maximum principle for solving diffusion problem with discontinuous

There are many unicellular tiny organisms which can self-propel collectively through non-Newtonian fluids by means of producing undulating deformation. Example includes nematodes, rod shaped bacteria and spermatozoa. Here we use Taylor’s swimming sheet model, with non-Newtonian fluid bounded with in a complex wavy walls of a two-dimensional channel. Oldroyd-4 constant fluid is approximated as cervical

This is a numerical study of convective heat transfer of the Water-Al2O3 nanofluid in an oval channel using two-phase mixture model. The channel is fitted with two rows of twisted conical strip inserts with various directions relative to each other leading to three different combinations of the mentioned inserts, namely inward Co-Conical inserts (CCI-inward), Counter-Conical inserts (CoCI), and outward

A multiple solutions finder method for steady Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) is designed combining the classical finite differences discretization approach and homotopy continuation with hyperspherical path-tracking. The proposed methodology was independently validated employing a reported multiple solutions ODE, and a designed non-linear 2-D PDE with

Angle-ply laminated plates have been examined for flexure analysis with multiquadric radial basis function (MQRBF) base meshfree method. A new higher order shear deformation theory (HSDT) representing the transverse shear strain function is proposed in the displacement field. Proposed theory satisfies the condition of continuity and differentiability and is accurate enough to predict the results of

This paper presents the interior penalty discontinuous Galerkin (IPDG) methods to solve the stationary Stokes equations with nonlinear damping term. The IPDG discrete schemes are established on general meshes. The corresponding consistency and stabilization of those schemes are proved. Subsequently, we analyze the existence, boundedness and uniqueness of the discrete solutions. Then the optimal error

We introduce a framework for spline spaces of hierarchical type, based on a parent–children relation, which is very convenient for the analysis as well as the implementation of adaptive isogeometric methods. This framework exploits the innate refinement by functions in the B-splines context, rather than by elements or cells, which is more natural in the finite element context. Furthermore, it entails

We extend the positive-definite and skew-Hermitian splitting (PSS) iterative method to EPSS (for extended PSS) for solving non-Hermitian positive-definite systems of linear equations. In this kind of extension, the shift matrix is replaced by a Hermitian positive-definite matrix Σ. It is proved that the method is unconditionally convergent. Then, the EPSS method with a 2 × 2-block diagonal matrix Σ

Based on the second-order cosine spectral differentiation matrix deduced in this paper, we present an efficient and high accurate numerical scheme for solving two-dimensional Schrödinger equation with Neumann boundary conditions. The Crank–Nicolson finite difference scheme is utilized in temporal discretization and the cosine spectral-collocation method is employed for the approximation of Laplacian

We present a new stabilization technique for multiscale convection–diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Péclet numbers. Our method is based on a constraint energy minimization idea and the discontinuous Petrov–Galerkin formulation. In particular, the test functions are constructed by minimizing an appropriate energy subject

In this study, a higher order compact (HOC) finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of Navier–Stokes (N–S) equations with the level set method. Unlike most of the existing methods for multiphase flows that solve the incompressible Navier–Stokes equations in the velocity–pressure form or

This study investigates viscoelastic fluid, which possesses both viscous and elastics properties, by employing a fractional derivative model to reveal the stress relaxation phenomenon with distance. The spatial-fractional derivative used in the momentum conservation equation is the Riemann–Liouville type derivative. Further, we use non-uniform boundary conditions subject to the boundary layer equations

This article investigates the bending response performances of the magneto–electro–elastic (MEE) laminated plates resting on the Winkler foundation or the elastic half-space subjected to a transverse mechanical loading .By assuming that the foundation is not electrically and magnetically conductive, the scaled boundary finite element method (SBFEM) based on the three-dimensional (3D) theory of elasticity

In this paper, we study an effective numerical method for the generalized Ginzburg–Landau equation (GLE). Based on the linearized Crank–Nicolson difference method in time and the standard Galerkin finite element method with bilinear element in space, the time-discrete and space–time discrete systems are both constructed. We focus on a rigorous analysis and consideration of unconditional superconvergence

The Partition of Unity Finite Element Method (PUFEM) has been widely used for the numerical simulation of the Helmholtz equation in different physical settings. In fact, it is a numerical pollution-free alternative method to the classical piecewise polynomial-based finite element methods. Taking into account a plane wave enrichment of the piecewise linear finite element method, the main goal of this

In this paper we propose a new projection method to solve both large-scale continuous-time matrix Riccati equations and differential matrix Riccati equations. The new approach projects the problem onto an extended block Krylov subspace and gets a low-dimensional equation. We use the block Golub–Kahan procedure to construct the orthonormal bases for the extended Krylov subspaces. For matrix Riccati

The Morse–Ingard equations of thermoacoustics Morse and Ingard (1986) are a system of coupled time-harmonic equations for the temperature and pressure of an excited gas. They form a critical aspect of modeling trace gas sensors. In this paper, we analyze a reformulation of the system that has a weaker coupling between the equations than the original form. We give a Gårding-type inequality for the system

We use the alternating direction method to simulate implicit dynamics. Our spatial discretization uses isogeometric analysis. Namely, we simulate a (hyperbolic) wave propagation problem in which we use tensor-product B-splines in space and an implicit time marching method to fully discretize the problem. We approximate our discrete operator as a Kronecker product of one-dimensional mass and stiffness

We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale Hr(Rn) and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale Hr(Rn)

Based on the block triangular splitting of the original coefficient matrix, a relaxed upper and lower triangular splitting (RULT) preconditioner for the linearized incompressible Navier–Stokes equations is considered. The convergence analysis of the RULT iteration method is presented. The spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed, respectively

deal.II is a state-of-the-art finite element library focused on generality, dimension-independent programming, parallelism, and extensibility. Herein, we outline its primary design considerations and its sophisticated features such as distributed meshes, hp-adaptivity, support for complex geometries, and matrix-free algorithms. But deal.II is more than just a software library: It is also a diverse

We present version 3 of the open-source simulator for flow and transport processes in porous media DuMux. DuMux is based on the modular C++ framework Dune (Distributed and Unified Numerics Environment) and is developed as a research code with a focus on modularity and reusability. We describe recent efforts in improving the transparency and efficiency of the development process and community-building

The discontinuous Petrov–Galerkin (DPG) method minimizes a residual in a non-standard norm. This paper shows that the minimization of this residual is equivalent to the minimization of a residual in a L2 norm. Since such residuals are well known from least squares finite element methods, this novel interpretation allows to extend results for least squares methods to the DPG method and vice versa. This

This paper applies a semi-analytical boundary collocation method, the singular boundary method (SBM), in conjunction with the dual reciprocity method (DRM) and Laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials (FGMs). In this study, transient heat conduction equation with Caputo time fractional derivative is considered to describe anomalous

This research focuses on performing multiphase solid/liquid/gas CFD simulations of a UASB reactor in order to obtain a validated model that provides a clearer understanding of the hydrodynamic behaviour of the three phases in UASB reactors. Eulerian–Eulerian, laminar, three-dimensional, multiphase simulations are carried out using Fluent 16.2. The liquid phase velocity and flow profile are validated

The problem of reconstructing the initial condition in the Neumann problem for linear parabolic equations with space-and-time-dependent coefficients from noisy observation of the solution on a part of the boundary is studied. Three variational methods are suggested for solving the problem: 1) the least squares method which aims at the minimizing the misfit between the observation on the boundary by

The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation

Computational simulations of transcranial direct current stimulation (tDCS) enable researchers and medical practitioners to investigate this form of neurostimulation with in silico experiments. For these computer-based simulations to be of practical use to the medical community, patient-specific head geometries and finely discretized computational grids must be used. As a result, solving the partial

This paper deals with the numerical computation and analysis for a class of two-dimensional time–space fractional convection–diffusion equations. An implicit difference scheme is derived for solving this class of equations. It is proved under some suitable conditions that the derived difference scheme is stable and convergent. Moreover, the convergence orders of the scheme in time and space are also

Herein, the solution of partial differential equations (PDEs) using spectral methods is developed for irregular domains, which preserves their accuracy. Previously, to solve these problems, the finite differences method or the embedded domains method was typically applied. The approach presented in this article can be used for any boundary described by a Jordan curve, and the solution behavior outside

Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the

Two powerful and versatile code packages (Agros Suite and Ārtap) are presented for design and optimization of technical devices and systems. Agros Suite represents an environment for numerical solution of systems consisting of partial differential equations (PDEs) of the second order by a higher-order finite element method with a lot of further advanced features such as full adaptivity and selected