In this paper we propose a new distribution of Gaussian points to compute the weak form integrals of the Generalized Finite Element Method (GFEM). The relevance of this new distribution is the possibility of evaluating the integrals of oscillatory functions inside the reference triangle in an alternate way. A simple scheme of relocation of the quadrature points allows to improve the efficacy of the

A convergence result is stated for the numerical solution of space-fractional diffusion problems. For the spatial discretization, an arbitrary family of finite elements can be used combined with the matrix transformation technique. The analysis covers the application of the implicit Euler method for time integration to ensure unconditional stability. The spatial convergence rate does not depend on

This paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data. With appropriately graded temporal grids, first-order temporal accuracy in the norm of L2(0,T;L2(Ω)) is derived by a new discrete Grönwall’s inequality, and then first-order temporal accuracy in the norm of L∞(0,T;L2(Ω)) is established for 1∕2<α<1. Finally, numerical results are

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution.

We revise the structure-preserving finite element method in [K. Hu, Y. MA and J. Xu. (2017) Stable finite element methods preserving ∇⋅B=0 exactly for MHD models. Numer. Math., 135, 371-396]. The revised method is semi-implicit in time-discretization. We prove the linearized scheme preserves the divergence free property for the magnetic field exactly at each time step. Further, we showed the linearized

A new stochastic control problem of a dam–reservoir system installed in a river is analyzed both mathematically and numerically. Water balance dynamics of the reservoir are piece-wise deterministic and are driven by a stochastic regime-switching inflow process. The system is controlled to balance among the operation purpose and the internal and downstream environmental conditions. Finding the optimal

In this article, we consider recovery-based a posteriori error estimators for the Virtual Element Method (VEM) and the Boundary Element Method based Finite Element Method (BEM-based FEM). Both methods are Galerkin methods on polygonal and polyhedral grids for the numerical solution of partial differential equations. Thus, they are highly flexible and particularly efficient in combination with adaptive

This paper focuses on the problem of image restoration under Cauchy noise. The variational method, which constructs the data fidelity term involving the Cauchy distribution by MAP estimator, has been proven to be a successful approach. In this paper, a nonlinear diffusion equation is proposed to deal with it. The main ingredients of the proposed equation are a gray level based diffusivity that estimates

An eigenvalue problem is introduced for the magnetohydrodynamic (MHD) Stokes equations describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. The solution of the eigenproblem is approximated by using a spectral collocation method that is based on vanishing the residual equation at the collocation points on the physical domain which

In Isogeometric Analysis, the computational domain is often described as multi-patch, where each patch is given by a tensor product spline/NURBS parametrization. In this work we propose a FETI-like solver where local inexact solvers exploit the tensor product structure at the patch level. To this purpose, we extend to the isogeometric framework the so-called All-Floating variant of FETI, that allows

We perform a followup computational study of the recently proposed space–time first order system least squares ( FOSLS ) method subject to constraints referred to as CFOSLS where we now combine it with the new capability we have developed, namely, parallel adaptive mesh refinement (AMR) in 4D. The AMR is needed to alleviate the high memory demand in the combined space time domain and also allows general

In this work, a flexible higher-order space–time adaptive finite element approximation of convection-dominated transport with coupled fluid flow is developed and studied. Convection-dominated transport is a challenging subproblem in poromechanics in which coupled transport with flow, chemical reaction and mechanical response in porous media is considered. Key ingredients are the arbitrary degree discontinuous

In this paper, we propose an explicit time-stepping scheme for the pattern formation in reaction–diffusion systems on evolving surfaces. The proposed numerical method is based on a simple discretization scheme of Laplace–Beltrami operator over triangulated surface. On the static and evolving domains, we perform various numerical experiments for effect of domain growth and pattern formations. The computational

The Flux Reconstruction approach is a recent high-order method which has been introduced for unsteady problems. Initial energy stability has been conducted for the advection problem, leading to the well know Energy Stable Flux Reconstruction (ESFR) scheme. Using the ESFR scheme, the energy stability proof has been extended for the advection–diffusion using the Local Discontinuous Galerkin (LDG) numerical

The coupled Darcy–Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolithic solution algorithms of the coupled Darcy–Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between

The present work documents the current state of development for our MATLAB/GNU Octave-based open source toolbox FESTUNG (Finite Element Simulation Toolbox for UNstructured Grids). The goal of this project is to design a user-friendly, research-oriented, yet computationally efficient software tool for solving partial differential equations (PDEs). Since the release of its first version, FESTUNG has

A distributed elliptic control problem with control constraints is considered, which is formulated as a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. Two discretization approaches with hp-finite elements are discussed: In the first discretization approach all variables (state, co-state

We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the method are tuned against available training data. In

A predictor of failure initiation in high-cycle fatigue was calibrated on the basis of uniaxial experimental data and its predictive performance tested against biaxial data. The procedures of solution verification, calibration, ranking, validation and uncertainty quantification were employed. A condition that defines the domain of the predictor is inferred from the results of validation experiments

In this paper, a least-squares virtual element method is presented for approximating the vector and scalar variables of second-order elliptic problems. The H(div)-conforming and scalar-conforming virtual elements are used to approximate the vector and scalar variables, respectively. The method allows the use of very general polygonal meshes and leads to a symmetric positive definite system. The optimal

Excessive synchronizations of neurons in the brain networks can be a reason for some episodic disorders such as epilepsy. In this paper, we simulate neural dynamic models and their synchronizations numerically by a new numerical algorithm. In order to overcome the complexity of the problem, a suitable combination of the Legendre spectral element method and operator splitting technique is proposed to

An adaptive BDDC (Balancing Domain Decomposition by Constraints) method is considered for three dimensional elliptic problems with coefficients of random variation and high contrast. For such model problems, a certain generalized eigenvalue problem on each subdomain interface is formed to select primal constraints adaptively. In three dimensions, eigenvalue problems are formed on each face or edge

In this work we introduce and analyze a new augmented fully-mixed formulation for the stationary Navier–Stokes/Darcy coupled problem. Our approach employs, on the free-fluid region, a technique previously applied to the stationary Navier–Stokes equations, which consists of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, together with the pressure. In

This paper concerns monolithic and splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE’s, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic

This paper is concerned with the study of wavelet approximation scheme based on Legendre and Chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations. The operational matrix for distributed order fractional differential operator is derived for Legendre and Chebyshev wavelets basis. Furthermore, the obtained operational matrix along with Gauss Legendre

This paper proposes an isogeometric boundary element (IGABEM) with the aim to solve the 2D steady-state heat conduction problems under spatially varying conductivity and internal heat source. The IGABEM boundary-domain integral equation is derived underlying the divergence theorem of Gauss and the Laplace equation. The non-uniform rational B-spline (NURBS) basis used in the CAD/CAE industries is applied

In this article, a fast time two-mesh (TT-M) finite element (FE) method for the two-dimensional space fractional Gray–Scott model is studied and discussed to get the numerical solutions effectively. The method mainly includes three steps: firstly, one uses an iterative method for solving the coupled nonlinear system on the time coarse grid; secondly, by an interpolation formula, one can get any coarse

Kelvin–Helmholtz instability (KHI) occurs at the interface of two fluids, in which the heavier fluid flows at the bottom. In the present numerical work, the KHI was analyzed by solving the modification of two-dimensional (2-D) incompressible Navier–Stokes equations. To capture the interface, the Cahn–Hilliard equation was implemented into the Navier–Stokes equations. The phenomena around the KHI of

Due to the multidisciplinary nature for the solution of magneto-electro-elastic (MEE) shell structures, developing a novel and accurate computational model is both essential and necessary for the practical engineering. The scaled boundary finite element method (SBFEM) is a semi-analytical technique in which only the surfaces or boundaries of the computational domain need to be discretized, while an

In this work, a high-order upwind compact finite-difference lattice Boltzmann method (UCDLBM) is developed to efficiently solve viscous incompressible flow problems. A fifth-order upwind compact difference scheme is adopted to discretize the spatial derivatives of the lattice Boltzmann equation, and the third-order total-variation-diminishing Runge–Kutta scheme is utilized for the discretization of

Following Muga and van der Zee (Muga and van der Zee, 2015), we generalize the standard Discontinuous Petrov–Galerkin (DPG) method, based on Hilbert spaces, to Banach spaces. Numerical experiments using model 1D convection-dominated diffusion problem are performed and compared with Hilbert setting. It is shown that Banach-based method gives solutions less susceptible to Gibbs phenomenon. h-adaptivity

We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements

In this work, we propose a telegraph coupled partial differential equation (TCPDE) based model for image restoration. The new framework interpolates between a couple of nonlinear telegraph equation and a parabolic equation. This proposed strategy can be applied to significantly preserve the oscillatory and texture pattern in an image, even in low signal-to-noise ratio (SNR). First, we prove that the

In this paper, we develop a cascadic multigrid asymptotic-preserving discrete ordinate discontinuous streamline diffusion scheme for radiative transfer equations (RTE) with multiple scalings. Our new method employs a simple and efficient cascadic multigrid method to the discretized RTE system as well as a diffusion synthetic acceleration technique as an efficient smoother to accelerate the convergence

We present a framework that relates preconditioning with a posteriori error estimates in finite element methods. In particular, we use standard tools in subspace correction methods to obtain reliable and efficient error estimators. As a simple example, we recover the classical residual error estimators for the second order elliptic equations.

We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a

A simple direct heating thermal immersed boundary-lattice Boltzmann method for buoyancy-driven incompressible flow with complex geometry boundaries is developed to simulate natural convection with curved boundary. In the newly developed method, both Dirichlet and Neumann boundary conditions in macroscopic equation are deduced into a novel mesoscopic discrete heat source. The mesoscopic discrete heat

In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm was already introduced in both the Brownian context and the Ornstein–Uhlenbeck context, that is for particular time-homogeneous diffusion processes. Here the aim is therefore to generalize this efficient numerical approach in order to obtain an approximation

We propose and analyze new stabilized time marching schemes for Phase Fields model such as Allen–Cahn and Cahn–Hillard equations, when discretized in space with high order finite differences compact schemes. The stabilization applies to semi-implicit schemes for which the linear part is simplified using sparse pre-conditioners. The new methods allow to significantly obtain a gain of CPU time. The numerical

A new combined discontinuous Galerkin method is proposed for compressible miscible displacement problem in porous media. Here, a splitting positive definite mixed finite element method is used for the pressure and Darcy velocity, while an interior penalty discontinuous Galerkin (IPDG) method is used for the transport equation. The stability and convergence of this algorithm are considered, and the

In the present work, a generalized finite difference method (GFDM), a meshless method based on Taylor-series approximations, is proposed to solve stationary 2D and 3D Stokes equations. To overcome the troublesome pressure oscillation in the Stokes problem, a new simple formulation of boundary condition for the Stokes problem is proposed. This numerical approach only adds a mixed boundary condition

This paper presents a new finite element solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of the classical bilinear or trilinear Lagrangian elements. It solves the primal variable displacement in the strain–div formulation and can handle both displacement and traction boundary conditions. It is a locking-free solver based on conforming finite elements. The solver

It is known that for exterior acoustics the domain-based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the Sommerfeld radiation condition. In this work, by using the polynomial and radial bases with specific node-selection schemes, the point interpolation method with the generalized gradient smoothing technique is employed to improve

We construct a general family of DPG Fortin operators for the exact energy spaces defined on a tetrahedral element.

This paper is devoted to a new formulation which couples the weak Galerkin method and the finite element method for approximating solutions of the equations of quasi-static poroelasticity which model flow through elastic porous media. It is assumed that the permeability of the elastic matrix depends nonlinearly on the dilatation of the porous medium. The steady-state version of the system is recast

In the spirit of the “Principle of Equipresence” introduced by Truesdell & Toupin, The Classical Field Theories (1960), we use the full version of the viscous stress tensor ν∇u+∇uT−23∇⋅uI which was originally derived for compressible flows, instead of the classical incompressible stress tensor ν∇u. (Note that, here ν is the dynamic viscosity coefficient, and u is the velocity field.) In our approach

In this work, phase-field modeling of hydraulic fractures in porous media is extended towards a Global–Local approach. Therein, the failure behavior is solely analyzed in a (small) local domain. In the surrounding medium, a simplified and linearized system of equations is solved. Both domains are coupled with Robin-type interface conditions. The fractures inside the local domain are allowed to propagate

A non-intrusive extension to the standard p-version of the finite element method is proposed. Meshes with hanging nodes are handled by adapting the reference elements so that the resulting discretisation is always conforming. The shape functions on these adaptive reference elements are not polynomials, but either harmonic extensions of the boundary restrictions of the standard shape functions or solutions

The cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics. In recent years, considerable attention has been devoted to distributed-order differential equations because they appear to be more effective for modeling complex processes. In this work, a finite difference/Legendre spectral method is presented for the numerical simulation of the two-dimensional

Significant developments in the field of additive manufacturing (AM) allowed the fabrication of complex microarchitectured components with varying porosity across different scales. However, due to the high complexity of this process, the final parts can exhibit significant variations in the nominal geometry. Computed tomographic images of 3D printed components provide extensive information about these

We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement

In this paper, lattice Boltzmann simulation for unsteady shock wave/boundary layer interaction in a shock tube is presented. This kind of problem, which contains the typical characteristics of supersonic viscous flows with shock wave and boundary layer, has rarely been well validated in the lattice Boltzmann community. This leads to few application of lattice Boltzmann method in supersonic flows with

When simulating complex physical phenomena such as aircraft icing or de-icing, several dedicated solvers often need to be strongly coupled. In this work, a non-overlapping Schwarz method is constructed with the unsteady simulation of de-icing as the targeted application. To do so, optimized coupling coefficients are first derived for the one dimensional unsteady heat equation with linear boundary conditions

We summarize some general ideas regarding approximation of mixed variational problems using saddle point reformulation. We consider the concepts of optimal and almost optimal (or α) test norm and provide estimates for the continuity and stability constants. A preconditioning strategy for solving the discrete mixed formulations is used in combination with the special test norms. We further provide a

This work aims at employing the discontinuous Galerkin (DG) methods for solving the nonstationary conduction–convection model, which is motivated by advantages of DG methods. The discontinuous element pair Pk-Pk−1-Pk in the space discretization and backward Euler scheme in time are applied, as well as an explicit upwind scheme based on Picard iteration is introduced for the coupled term. Existence

The classical dual mixed finite element method for flow simulations is based on H(div,Ω) conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in L2(Ω) for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide

We develop and analyze an ultraweak formulation of linear PDEs in nondivergence form where the coefficients satisfy the Cordes condition. Based on the ultraweak formulation we propose discontinuous Petrov–Galerkin (DPG) methods. We investigate Fortin operators for the fully discrete schemes and provide a posteriori estimators for the methods under consideration. Numerical experiments are presented

This work is a first taste of using null-space technique to deal with a basic model of unsteady partial differential equations (PDEs), i.e., the unsteady Stokes equations. Any inf–sup stable semi-discretization of the unsteady Stokes equations yields a system of differential–algebraic equations (DAEs), i.e., the unsteady discrete Stokes equations. The exact solution to the unsteady discrete Stokes

In this paper, we propose a fast algorithm for the variable-order (VO) Caputo fractional derivative based on a shifted binary block partition and uniform polynomial approximations of degree r. Compared with the general direct method, the proposed algorithm can reduce the memory requirement from O(n) to O(rlogn) storage and the complexity from O(n2) to O(rnlogn) operations, where n is the number of

This article introduces a new family of quadrature rules for integrating smooth functions on 4-dimensional simplex elements (i.e. pentatopes). These quadrature rules have 1, 5, 15, 35, 70, and 126 points, and are capable of exactly integrating polynomials of degrees 1, 2, 3, 5, 6, and 8, respectively. One main advantage of these rules, is that they have a ‘pentatopic number’ of points, which means