A high order discontinuous Galerkin method with Lagrange multiplier (DGLM) in space combined with discontinuous Galerkin (DG) method in time (DG-DGLM) [32] is numerically investigated for the approximation of the solution to the system of hyperbolic conservation laws. Computation is done in element by element fashion. P0 time and space subcell limiting processes are applied to resolve the shocks. It

The space distributed-order constitution relationship is firstly formulated to study the flow and heat transfer in the boundary layer. The distributed-order boundary layer equations are derived and approximated by a multi-space fractional order ones, which are then solved by the implicit difference schemes. By comparing the analytical solutions of special boundary conditions, the validity of the present

Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric M-Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation AX+XB=C with complex symmetric coefficient matrices

This paper is devoted to a rigorous mathematical foundation for the convergence properties of the strain-smoothed element (SSE) method. The SSE method has demonstrated improved convergence behaviors compared to other strain smoothing methods through various numerical examples; however, there has been no theoretical evidence for the convergence behavior. A unique feature of the SSE method is the construction

The fictitious boundary surrounding the domain is required in the implementation of the method of fundamental solutions (MFS). A similar approach of taking a portion of the centres of radial basis functions (RBFs) outside the domain in the context of the method of particular solutions (MPS) is proposed to further improve the performance of a two-step MPS-MFS method. Since the shape parameter of RBFs

We design and analyze a posteriori error estimators for the Stokes system with singular sources in suitable W1,p×Lp spaces. We consider classical low-order inf-sup stable and stabilized finite element discretizations. We prove, in two and three dimensional Lipschitz, but not necessarily convex polytopal domains, that the devised error estimators are reliable and locally efficient. On the basis of the

In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which have favorable scalings with respect to the diffusion parameter when compared with

A direct finite element solver is considered for symmetric sparse matrices arising from the application of the finite element method to problems of structural mechanics and solid mechanics. The solver is based on the block Cholesky method generalized to indefinite matrices. A distinctive feature of the proposed method from other known approaches is the original algorithm for parallelizing the factorization

We observe a dramatic lack of robustness of the DPG method when solving problems on large domains and where stability is based on a Poincaré-type inequality. We show how robustness can be re-established by using appropriately scaled test norms. As model cases we study the Poisson problem and the Kirchhoff–Love plate bending model, and also include fully discrete variants where optimal test functions

In this paper, the discontinuous Galerkin method (DGM) of nonconforming Wilson element is studied for the semi-linear parabolic problem. The global superconvergence with respect to the mesh size are derived in the modified H1-norm for the semi-discrete scheme and two fully discrete schemes, in which the usual extrapolation and interpolation post-processing approaches are not involved, and the error

Two efficient numerical schemes based on L1 formula and Finite-Difference Time-Domain (FDTD) method are constructed for Maxwell's equations in a Cole-Cole dispersive medium. The temporal discretizations are built upon the leap-frog method and Crank-Nicolson method, respectively. We carry out the energy stability and error analysis rigorously by the energy method. Both schemes have been proved convergence

This work is concerned with a mathematical study of backward problem for time-space fractional diffusion equations associated with the observed data measured in Hilbert scales. Transforming the original problem into an operator equation, we investigate the existence, the uniqueness and the instability for the problem. In order to overcome the ill-posedness of the problem, we apply a modified version

In this paper, we develop a partitioned scheme with multiple-time-step technique for the nonstationary dual-porosity-Stokes model. The proposed partitioned scheme allows different time steps in the different subdomains of the dual-porosity-Stokes model: conduits/macrofractures, microfractures, and matrix. Under a time step restriction which depends on the physical parameters of the model, we derived

This article discusses nonconforming finite element methods for convex minimization problems and systematically derives dual mixed formulations. Duality relations lead to simple error estimates that avoid an explicit treatment of nonconformity errors. A reconstruction formula provides the discrete solution of the dual problem via a simple postprocessing procedure which implies a strong duality relation

We propose a class of numerical schemes for nonlocal HJB variational inequalities (HJBVIs) with monotone nonlinearities arising from mixed optimal stopping and control of processes with infinite activity jumps, where the objective is specified by a monotone recursive preference. The solution and free boundary of the HJBVI are constructed from a sequence of penalized equations, for which the penalization

In this paper, superconvergence properties are presented for the nonlinear Benjamin-Bona-Mahony (BBM) equation with a mixed finite element method (MFEM). We propose an Euler scheme combined with the artificial Douglas-Dupont regularization terms, which guarantee the stability of the numerical scheme. Splitting technique is utilized to get rid of the ratio between the time step τ and the subdivision

We deal with the goal-oriented error estimates and mesh adaptation for nonlinear partial differential equations. The setting of the adjoint problem and the resulting estimates are not based on a differentiation of the primal problem but on a suitable linearization which guarantees the adjoint consistency of the numerical scheme. Furthermore, we develop an efficient adaptive algorithm which balances

With the objective of studying fluid-structure interaction (FSI) applications with sophisticated structural behavior, this work studies an approach that makes room for the use of complex structural models without penalizing the global computational time. The proposed approach is based on an algorithm that, simultaneously solving two subdomains, allows an asynchronous update of the boundary conditions

Free boundary problems deal with systems of partial differential equations, where the domain boundaries are apriori unknown. Due to this special characteristic, it is challenging to solve free boundary problems either theoretically or numerically. In this paper, we develop a novel approach for solving a modified Hele–Shaw problem based on the neural network discretization. The existence of the numerical

In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment

Anisotropic diffusion filtering is a widely used partial differential equation (PDE) based technique, which works effectively for removal of noise and preserving edges. This technique is highly dependent on diffusion coefficient and threshold parameter. In this work, we propose a new diffusion coefficient and image dependent threshold parameter which has a faster rate of convergence than those found

This article addressees the dynamics of fluid conveying tinny particles and Coriolis force effects on transient rotational flow toward a continuously stretching sheet. Tiny particles are considered due to their unusual characteristics like extraordinary thermal conductivity, which are significant in advanced nanotechnology, heat exchangers, material sciences, and electronics. The main objective of

Overlapping Additive Schwarz (OAS) preconditioners are here constructed for isogeometric collocation discretizations of the system of linear elasticity in both two and three space dimensions. Isogeometric collocation methods are recent variants of isogeometric analysis based on the numerical approximation of the strong form of partial differential equations at appropriate collocation points. Numerical

In this paper, two linearized Galerkin finite element methods, which are based on the L1 approximation and the WSGD operator, respectively, are proposed to solve the nonlinear fractional Rayleigh-Stokes problem. In order to obtain the unconditionally optimal error estimate, we firstly introduce a time-discrete elliptic equation, and derive the unconditional error estimate between the exact solution

In this paper, we study the simplest and the most standard adaptive finite element method for the second-order nonmonotone quasi-linear elliptic problems with the exact solution u∈H01+α(Ω),α≥12. The adaptive algorithm is based on the residual-based a posteriori error estimators and Dörfler’s marking strategy. We prove the convergence and quasi-optimality of the adaptive finite element method when the

In this paper, an equal-order hybridized discontinuous Galerkin (HDG) method with a small pressure penalty parameter for the Stokes equations is analyzed. When the pressure penalty parameter γ tends to 0 (γ>0), the velocity approximation tends to be H(div)-conforming and exactly divergence-free, and the velocity error bound tends to be pressure-robust. However, taking the value of γ too small will

In this article we analyze the linear parabolic partial differential equation with a stochastic domain deformation. In particular, we concentrate on the problem of numerically approximating the statistical moments of a given Quantity of Interest (QoI). The geometry is assumed to be random. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit

We consider a singularly perturbed convection-diffusion boundary value problem whose solution contains exponential and characteristic boundary layers. The problem is numerically solved by the FEM and SDFEM method with bilinear elements on a graded mesh. For the FEM we prove almost uniform convergence and superconvergence. The use of a graded mesh allows for the SDFEM to yield almost uniform estimates

Distributed-order fractional differential equations, where the differential order is distributed over a range of values rather than being just a fixed value as it is in the classical differential equations, offer a powerful tool to describe multi-physics phenomena. In this article, we develop and analyze an efficient finite difference/generalized Hermite spectral method for the distributed-order time-fractional

A novel near-wall modeling for large-eddy simulation based on the lattice Boltzmann method is proposed. An existing near-wall modeling based on the reconstruction of distribution functions is further developed to calculate high Reynolds number wall-bounded turbulent flow on a non-body-fitted Cartesian grid. The following three ideas are introduced into the existing reconstruction model: (1) the introduction

We present a space–time least-squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space–time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have

In this article, the applicability of the cumulant lattice Boltzmann method (CuLBM) to simulate boundary layer flow above rough surfaces is investigated. The fluid is assumed to be incompressible and Newtonian and its flow is investigated under isothermal conditions above surfaces with small obstacles. The results obtained by the CuLBM are confronted with the results of a reference finite difference-based

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion–convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a

We present in this work a unified framework for elliptic variational inequalities that gathers several problems in contact mechanics like the unilateral contact of one or two membranes or the Signorini problem. We study a family of Galerkin numerical schemes that discretize this framework. We prove the well-posedness of the discrete problem and we show that it is equivalent to a saddle-point mixed

Understanding the physics of droplet collisions is of crucial importance in many industrial processes such as inkjet printing, spray cooling, and spray combustion. In this paper, a modified multiple-relaxation-time pseudopotential multiphase lattice Boltzmann model is proposed to investigate binary droplet collision with different collision angles. The surface tension can be adjusted independently

This work aims to build a bridge between probability methods and finite element methods. It starts with considering probability distributions supported in an interval [a,b], which incorporate the traditional probability distributions defined on the whole real space as limit cases on the one hand, lead to a type of spherical probability models with wide potential applications on the other hand. This

The application of the TR-BDF2 method to second order problems typical of structural mechanics and seismic engineering is discussed. A reformulation of this method is presented, that only requires the solution of algebraic systems of size equal to the number of displacement degrees of freedom. A linear analysis and numerical experiments on relevant benchmarks show that the TR-BDF2 method is superior

For the two dimensional anisotropic diffusion problems, we postprocess the k-th order finite element solution to satisfy the local conservation law on a certain dual mesh. The postprocessing technique is independently implemented by solving a 3-by-3 and 4-by-4 local linear algebraic system on each triangular and quadrilateral element, respectively. For any full anisotropic diffusion tensors, and arbitrary

The paper studies an Allen–Cahn-type equation defined on a time-dependent surface as a model of phase separation with order–disorder transition in a thin material layer. By a formal inner–outer expansion, it is shown that the limiting behavior of the solution is a geodesic mean curvature type flow in reference coordinates. A geometrically unfitted finite element method, known as a trace FEM, is considered

The empirical observations show that the constant elasticity of variance (CEV) model is a practical approach to capture the implied volatility smile phenomenon. Also, due to the appearance of long-range dependence (or long memory effect) in stock returns or volatility, the fractional Black–Scholes models became particularly important in finance. We will generalize the CEV model to a fractional-CEV

This work is devoted to the development of a dissipative spatial discretization of a phase field model describing the dynamics of a multicomponent multiphase isothermal viscous compressible fluid with a resolved interface and effects associated with it (surface tension, wetting). Mass densities of mixture components are used as order parameters. The total Helmholtz free energy incorporates wall free

The corners of non-smooth yield surfaces, e.g., Mohr–Coulomb or Hoek–Brown criteria, often cause problems in numerical applications due to the gradient discontinuities, which boil down to the abrupt order interchange of two principal stresses. Nevertheless, when the stress components cross the corners, the principal stresses, which are smoothly dependent on the components of stress tensor, can be smoothly

In this paper, an effective linearized element-free Galerkin (EFG) method is developed for the numerical solution of the complex Ginzburg–Landau (GL) equation. To deal with the time derivative and the nonlinear term of the GL equation, an explicit linearized procedure is presented. The unconditional stability and the error estimate of the procedure are analyzed. Then, a stabilized EFG method is proposed

A Coiflet-type wavelet-homotopy approach is implemented to investigate the large deformation of a circular plate resting on different nonlinear foundations with various boundary parameters. A parameterized and continuous boundary model for circular plate with various constraints of rotation and translation has been proposed overlooked in previous studies. Parameterized wavelet approximation of arbitrary

A general method for the derivation of equivalent finite difference equations (EFDEs) and subsequent equivalent partial differential equations (EPDEs) is presented for a general matrix lattice Boltzmann method (LBM). The method can be used for both the advection diffusion equations and Navier–Stokes equations in all dimensions. In principle, the EFDEs are derived using a recurrence formula. A computational

In this paper, we obtain guaranteed lower bounds for eigenvalues of two spectral problems arising in fluid mechanics by using the min–max principles of weak form that can be derived by the principles of operator forms. These two problems are the Laplace model for fluid–solid vibrations and the sloshing problem. We deal with the positive semi-definiteness of the associated bilinear forms by adding some

In this paper, based on two-grid discretization, a new local and parallel finite element method is proposed and investigated for the mixed Navier–Stokes–Darcy problem. Compared with the classical local and parallel finite element method, the main features of our method are: (1) partition of unity method is considered to generate local subproblems; (2) global continuous approximations are achieved.

A mixture model to take into account the flow of small carbon dioxide bubbles dissolved in a liquid is presented. The model describes the evolution of the velocity fields (mixture and gas), the pressure and the volume fraction of gas. The system of equations is derived from mass and momentum conservation of the mixture and gas. Well-posedness is proved for a simplified problem when the volume fraction

The eigenvalue buckling responses of smart carbon nanotube-reinforced hybrid composite shell structure are analyzed under the influence of uniform thermal loading using a multiscale material model. The hybrid nanotube shell structural model is formulated mathematically via a cubic-order shear deformation theory introducing the material nonlinearity due to the shape memory alloy fiber. Additionally

In this paper, a novel system of Maxwell–Schrödinger equations with nonlocal effect in metamaterials is derived from the Drude model, hydrodynamical model and Schrödinger equation. A leap-frog finite element scheme, which can be solved one by one efficiently, is constructed by presenting a group of initial values. This scheme is proved to be stable conditionally in energy norm. It is confirmed that

In this paper, a family of linear structure-preserving (energy conservation) schemes with second-order accuracy in the time direction is developed to numerically solve the undamped sine–Gordon equation. To be specific, first, transformation of the undamped sine–Gordon system into an equivalent new system is made by introducing an improved scalar auxiliary variable (SAV), and generalization of the conservative

A flexible framework for shape optimisation is presented for incompressible Newtonian fluids using lattice Boltzmann methods. It is proposed to solve optimisation problems using line search methods, with design sensitivities obtained through forward propagation automatic differentiation. The underlying fluid flow problems are modelled by homogenised lattice Boltzmann methods, wherein permeability is

This paper analyzes the well-known L1 scheme for fractional wave equations with nonsmooth data. A new stability estimate and temporal accuracy O(τ3−α) are obtained. In addition, a modified L1 scheme is proposed, and its stability and temporal accuracy O(τ2) are also derived under nonsmooth data. The convergence of the two schemes in the inhomogeneous case are also established. Finally, numerical experiments

Matrix double splitting iterations are simple in implementation while solving real non-singular (rectangular) linear systems. In this paper, we present two Alternating Double Splitting (ADS) schemes formulated by two double splittings and then alternating the respective iterations. The convergence conditions are then discussed along with comparative analysis. The set of double splittings used in each

A well-defined grid line-based immersed boundary method is presented for efficient and accurate simulation of unsteady, incompressible flow using non-body conformal, Cartesian grids. Near the fluid–solid interface, the spatial discretization of Navier–Stokes equations is modified to enforce desired boundary conditions in a well-defined manner. Desired modifications can be stably derived using a one-dimensional

High dimensional conservative spatial distributed-order fractional diffusion equation is discretized by midpoint quadrature rule, Crank–Nicolson method, and a finite volume approximation, with alternating direction implicit scheme. The resulting scheme is shown to be consistent and unconditionally stable, hence convergent with order 3−α, where α is the maximum of the involving fractional orders. Moreover

In this study, a modified multilevel fast multipole algorithm is constructed for investigating large-scale particle systems. The algorithm expands the number of levels of the modified dual-level fast multipole algorithm from dual-level grids to multipole levels by a layer-by-layer correction and recursive calculation. The linear equations on coarse grid are recursively solved by a two-level grid. The

We analyze a nonlinear degenerate parabolic problem whose diffusion coefficient is the Heaviside function of the distance of the solution itself from a given target function. We show that this model behaves as an evolutive variational inequality having the target as an obstacle: under suitable hypotheses, starting from an initial state above the target the solution evolves in time towards an asymptotic

We develop a first- and second-order finite-volume scheme to solve gradient flow equations with non-homogeneous properties, obtained in the framework of dynamical-density functional theory. The scheme takes advantage of an upwind approach for the space discretization to ensure positivity of the density under a CFL condition and decay of the discrete free energy. Our computational approach is used to

In this paper, we study and analyse Crank–Nicolson (CN) temporal discretization with certain spatial difference schemes for one- and two-dimensional two-sided space-fractional diffusion equations (TSFDEs) with variable diffusion coefficients. The stability and convergence of the resulting discretization linear systems for TSFDEs with variable diffusion coefficients are proven by a new technique. That