In this paper, we aim to develop a thermodynamically consistent diffuse interface model for two-phase multicomponent fluids with partial miscibility based on the second law of thermodynamics. The resulting governing equations consist of the Allen-Cahn equation for the order parameter representing the phase volume fraction, the Cahn-Hilliard equation for the molar fraction of each component in the mixture

For a singular integral equation on an interval of the real line, we study the behavior of the error of a delta-delta discretization. We show that the convergence is non-uniform, between order O(h2) in the interior of the interval and a boundary layer where the consistency error does not tend to zero.

In this paper, we propose a new discontinuous sensor and a finite difference hybrid unequal-sized weighted essentially non-oscillatory (WENO) scheme with fifth-order accuracy for solving shallow water equations with or without source terms. The developed discontinuous sensor is directly designed based on the highest degree polynomial obtained from the five-point stencil in the unequal-sized WENO procedures

Parametric time-dependent systems are of a crucial importance in modeling real phenomena, often characterized by nonlinear behaviours too. Those solutions are typically difficult to generalize in a sufficiently wide parameter space while counting on limited computational resources available. As such, we present a general two-stage deep learning framework able to perform that generalization with low

In this paper, we consider the discretization of the two-dimensional stationary Stokes equation by Crouzeix-Raviart elements for the velocity of polynomial order k≥1 on conforming triangulations and discontinuous pressure approximations of order k−1. We will bound the inf-sup constant from below independent of the mesh size and show that it depends only logarithmically on k. Our assumptions on the

For the numerical simulation of time-dependent problems, recent works suggest the use of a time marching scheme based on a tensorial decomposition of the time axis. This time-separated representation is straightforwardly introduced in the framework of the Proper Generalized Decomposition (PGD). The time coordinate is transformed into a multi-dimensional time through new separated coordinates, the micro

Finite elements are allowed to be of a shape suitable for the specific problem. This choice defines thereafter the accuracy of the approximated solution. Moreover, flexible element shapes allow for the construction of an arbitrary domain topology. Polygon meshes are a common representation of the domain that cover any choice of the finite element shape. Being an alternative tool for modeling and analysis

Many stable and convergent numerical methods are not pressure robust for an incompressible flow problem, due to the relaxation of a discrete divergence constraint called poor mass conservation. As a result, the velocity error is polluted by the pressure error, particularly when the fluid viscosity is relatively small and pressure is comparably large. A grad-div stabilization is an efficient remedy

In this article, we propose a linearized fully-discrete scheme for solving a time fractional nonlocal diffusion-wave equation of Kirchhoff type. The scheme is established by using the finite element method in space and the L1 scheme in time. We derive the α-robust a priori bound and a priori error estimate for the fully-discrete solution in L∞(H01(Ω)) norm, where α∈(1,2) is the order of time fractional

Next-generation lithium-ion batteries with silicon anodes have positive characteristics due to higher energy densities compared to state-of-the-art graphite anodes. However, the large volume expansion of silicon anodes can cause high mechanical stresses, especially if the battery active particle cannot expand freely. In this article, a thermodynamically consistent continuum model for coupling chemical

Consider the quadratic finite element method over a regular family of uniform tetrahedral partitions for the three-dimensional Poisson equation. Utilizing properties of the bubble function and estimates for discrete Green's function, we derive a pointwise superconvergent estimate for the finite element approximation in the sense of L∞-norm. Furthermore, the L2-norm superconvergent estimate is also

In this study, we introduce a novel weighted essentially non-oscillatory (WENO) conservative finite-difference scheme that improves the performance of the known third-order WENO methods. To approximate sharp gradients and high oscillations more accurately, we incorporate an interpolation method using a set of exponential (or trigonometric) polynomials with an internal shape parameter. In particular

In this article, we use space-time continuous Galerkin (STCG) method to find the numerical solution for two-dimensional (2D) Burgers' equation. The STCG method differs from conventional finite element methods, both the spatial and temporal variables are discretized by finite element method, thus it can easily obtain the high order accuracy in both time and space directions and the corresponding discrete

Uncertainty quantification (UQ) tasks, such as sensitivity analysis and parameter estimation, entail a huge computational complexity when dealing with input-output maps involving the solution of nonlinear differential problems, because of the need to query expensive numerical solvers repeatedly. Projection-based reduced order models (ROMs), such as the Galerkin-reduced basis (RB) method, have been

In this work, a locking-free nonconforming virtual element method for the three-field Biot's model (the displacements, Darcy velocity and pore pressure) is developed and analyzed. We solve the model as: nonconforming VEM for the displacements, RT-like VEM for Darcy velocity and pressure. We show the well-posedness and the optimal error estimates of the discrete schemes. The numerical experiments support

In this paper, we propose new basis functions defined on curved sides or faces of curvilinear elements (polygons or polyhedrons with curved sides or faces) for the weak Galerkin finite element method. Those basis functions are constructed by collecting linearly independent traces of polynomials on the curved sides/faces. We then analyze the modified weak Galerkin method for the elliptic equation and

We consider an ultra-weak first order system discretization of the Helmholtz equation. When employing the optimal test norm, the ‘ideal’ method yields the best approximation to the pair of the Helmholtz solution and its scaled gradient w.r.t. the norm on L2(Ω)×L2(Ω)d from the selected finite element trial space. On convex polygons, the ‘practical’, implementable method is shown to be pollution-free

In this paper, we consider output tracking and disturbance rejection for a 1-D wave equation with in-domain feedback/recirculation of an intermediate point velocity. The performance output is an intermediate point displacement and the control matches the general disturbance. We first use the active disturbance rejection control method to design a disturbance estimator to estimate the disturbance. Then

In this paper, we develop a high-order finite difference scheme for the two-dimensional time-fractional reaction-diffusion equation with variably diffusion coefficient, in which the non-uniform L1 time stepping method on graded mesh is utilized for temporal discretization to compensate for the possible temporal accuracy lost caused by the initial weak singularity, and by introducing a flux variable

We develop a temporally second-order stabilizer-free weak Galerkin (SFWG) finite element method with unequal time steps for reaction-subdiffusion equation in multiple space dimensions. In consideration of initial singularity of the solution, we prove a sharp error estimate on nonuniform time steps by two tools: discrete fractional Grönwall inequality and error convolution structure analysis. Furthermore

This paper presents an alternative well-balanced and energy stable method for the system of non-homogeneous Shallow Water Equations (SWE) on unstructured grids based on the residual distribution (RD) approach. The motivation of the work is based on the grid insensitivity of RD method. The newly proposed method has a positive first order part, a linearity-preserving second order scheme as well as a

In this paper, we concentrate on design a bilateral preconditioning for all-at-once system from multidimensional time-space non-local evolution equations with a weakly singular kernel. Firstly, we propose an implicit difference scheme for this equation by employing an L2-type formula. Then, we prove the stability and the convergence of the proposed scheme. Based on such a numerical scheme, an L2-type

In order to make good use of the inherent advantages of the FEM and the BEM formulations, an iterative decomposition coupling algorithm between time-domain FEM and time-domain BEM with independent spatial discretization on the interface is proposed for elastodynamic problems. In the coupling process, based on the conditions of the displacement continuity and the force equilibrium on the interface,

This paper aims to propose positivity-preserving nonstaggered central difference schemes solving the two-layer shallow water equations with a nonflat bottom topography and geometric channels. One key step is to discretize the bed slope source term and the nonconservative product term due to the exchange of the momentum based on defining auxiliary variables. The second-order accuracy in space can be

A coefficient identification problem (CIP) for a system of one-dimensional advection-reaction equations using boundary data is considered. The advection-reaction equations are used to describe the transportation of pollutants in rivers or streams. Stability for the considered CIP is proved using global Carleman estimates. The CIP is solved using the least-squares approach accompanied with the adjoint

A learning technique for finite horizon optimal control problems and its approximation based on polynomials is analyzed. It allows to circumvent, in part, the curse dimensionality which is involved when the feedback law is constructed by using the Hamilton-Jacobi-Bellman (HJB) equation. The convergence of the method is analyzed, while paying special attention to avoid the use of a global Lipschitz

We propose a single-step simplified lattice Boltzmann algorithm capable of performing magnetohydrodynamic (MHD) flow simulations in pipes for very small values of magnetic Reynolds numbers Rm. In some previous works, most lattice Boltzmann simulations are performed with values of Rm close to the Reynolds numbers for flows in simplified rectangular geometries. One of the reasons is the limitation of

This paper presents an efficient parallel computing strategy to solve large-scale structural vibration problems. The proposed approach utilises a novel direct method that operates using simplex-shaped space-time finite elements and allows for the direct decoupling of variables during the assembly of global matrices. The method uses consistent stiffness, inertia and damping matrices and deals with non-symmetric

This paper presents a scalable multigrid preconditioner targeting large-scale systems arising from discontinuous Petrov–Galerkin (DPG) discretizations of high-frequency wave operators. This work is built on previously developed multigrid preconditioning techniques of Petrides and Demkowicz (Comput. Math. Appl. 87 (2021) pp. 12–26) and extends the convergence results from O(107) degrees of freedom (DOFs)

In the present work, we developed the Neural Networks (NNs) for identifying unknown surface shape of inner wall in the two-dimensional pipeline based on the temperature at uniformly distributed measuring points. The steady-state governing equation is transformed into the anisotropic heat conduction equations, and the irregularly shaped inner boundary is identified by the estimation of circumferential

In this paper, we apply the ideas of modulus methods to the problem of non-negatively constrained image restoration based on total variation (TV) regularization and present a modulus iteration method for this problem. The proposed method has the potential to effectively handle non-negative constraint for image restoration. We prove the convergence of the proposed method. Experimental results show that

We present a spacetime DG method for 1D spatial domains and three linear hyperbolic, damped hyperbolic, and parabolic PDEs. The latter two correspond to Maxwell-Cattaneo-Vernotte (MCV) and Fourier heat conduction problems. The method is called the tent-pitcher spacetime DG method (tpSDG) due to its resemblance to the causal spacetime DG method (cSDG) wherein the solution advances in time by pitching

In order to establish the dynamic equations of the rigid-flexible coupled robotic mechanisms with large beam-deformations over the horizontal plane, a thorough modeling technique founded on the virtual work concept has been proposed. Based on the transformed full-actuated model, the predefined-time robust sliding mode control strategy is developed to track the prescribed angular positions of the rigid-flexible

The equilibrium radiation diffusion equation has been widely used in astrophysics, inertial confinement fusion and others. Since the simulation domain consists of many complicated domains and the material properties in each domain are different, the diffusion coefficient usually has a strong discontinuity at the interface. Because the equilibrium radiation diffusion equation is often built on complicated

The purpose of this article is to study the convergence of a low order finite element approximation for a natural convection problem. We prove that the discretization based on P1 polynomials for every variable (velocity, pressure and temperature) is well-posed if used with a penalty term in the divergence equation, to compensate the loss of an inf-sup condition. With mild assumptions on the pressure

In this paper, a Legendre-tau Chebyshev collocation spectral element method is developed for solving Maxwell's equations with material interfaces in two dimensions. The transverse magnetic mode is considered mainly. The developed scheme treats the interface conditions in a way like the natural boundary condition based on a reasonable weak formulation, which makes the numerical solution retain the original

Weak Galerkin finite element methods (WG-FEMs) for H(curl;Ω) and H(curl,div;Ω)-elliptic problems are investigated in this paper. The WG method as applied to curl-curl and grad-div problems uses two operators: discrete weak curl and discrete weak divergence, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing

We discuss goal-oriented adaptivity in the frame of conforming finite element methods and plain convergence of the related a posteriori error estimator for different general marking strategies. We present an abstract analysis for two different settings. First, we consider problems where a local discrete efficiency estimate holds. Second, we show plain convergence in a setting that relies only on structural

Based on the traditional scalar auxiliary variable (SAV) method, the exponential SAV (ESAV) method, and the Lagrange multiplier method, three efficient energy stable schemes are proposed to solve the Klein-Gordon-Zakharov equations. All proposed schemes lead to linear equations with constant coefficients to be solved in each time step. The first two schemes are proved to preserve two different modified

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth multi-patch representation of their finite boundary surface. The discretization spaces are formed by C0 inter-patch continuous functional spaces whose restriction to a patch simplifies to the span of tensor product B-splines composed

In this paper we present an immersed weak Galerkin method for solving second-order elliptic interface problems on polygonal meshes, where the meshes do not need to be aligned with the interface. The discrete space consists of constants on each edge and broken linear polynomials satisfying the interface conditions in each element. For triangular meshes, such broken linear polynomials coincide with the

A type of artificial boundary layer for the numerical simulation of wave propagation, which is named the gradient viscoelastic (GV) boundary layer, is proposed in this study. The setting of the GV boundary layer was established according to the propagation behavior of the longitudinal wave in the elastic GV rod, the analytical solution of the longitudinal wave propagation in the GV rod was obtained

In this paper, we propose a simple and fast shape transformation method. This method is based on the Allen-Cahn (AC) partial differential equation and uses the edge stop function to constrain evolution. We use the operator splitting method to control the equation for splitting, and use the explicit Euler's method to solve the discrete equation. Based on the source shape and the target shape, our phase-field

An efficient parallel finite element method is introduced for solving the steady-state Smagorinsky model in which a fully overlapping domain decomposition is considered for parallelization. The crucial idea of the method is to utilize a locally refined multiscale mesh that is fine around its own subdomain and coarse elsewhere to calculate a local finite element solution. On the basis of an existing

In this paper, we present an immersed finite element (IFE) method for solving the elastodynamics interface problems on interface-unfitted meshes. For spatial discretization, we use vector-valued P1 and Q1 IFE spaces. We establish some important properties of these IFE spaces, such as inverse inequalities, which will be crucial in the error analysis. For temporal discretization, both the semi-discrete

In this paper, we propose a two-level additive Schwarz preconditioner for the Nitsche extended finite element discretization of elliptic interface problems. The intergrid transfer operators between the coarse mesh and the fine mesh spaces are constructed and a stable space decomposition is given. It is proved that the condition number of the preconditioned system is bounded by C(1+Hδ+Hh), where H and

Unsteady oblique stagnation point flow, heat and mass transfer of generalized Oldroyd-B fluid over an oscillating plate are investigated. The upper-converted derivative is introduced to the constitutive equation of fractional Oldroyd-B fluid. The terms of pressure are inventively solved by means of the momentum equation far from the plate. Furthermore, fractional Cattaneo–Christov double diffusion

In this paper we present a nonoverlapping Robin-type multi-domain decomposition method based on stabilized Q1–P0 mixed approximation (RMDD-Q1P0) for incompressible flow problems. The global Stokes and Navier-Stokes equations are decomposed into a series of local problems through Robin-type domain decomposition, and local problems are solved through the local jump stabilized Q1–P0 approximation. The

We present an anisotropic hp-mesh adaptation strategy using a continuous mesh model for discontinuous Petrov-Galerkin (DPG) finite element schemes with optimal test functions, extending our previous work [1] on h-adaptation. The proposed strategy utilizes the built-in residual-based error estimator of the DPG discretization to compute both the polynomial distribution and the anisotropy of the mesh

In this article, we investigate an inverse and ill-posed problem to simultaneously reconstruct the space-dependent source and initial value associated with a two-dimensional heat conduction equation based on the additional temperature data. An existence and uniqueness theorem is deduced by the contraction mapping principle. To obtain stable approximate solutions, we propose a fractional Landweber iteration

Approximating differential operators defined on two-dimensional surfaces is an important problem that arises in many areas of science and engineering. Over the past ten years, localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have been shown to be effective for this task as they can give high orders of accuracy at low computational

In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential operator is realized by automatic differentiation. The eigenfunction of the eigenvalue problem is learned by the neural network and the iterative algorithms are implemented

We present a unified framework for goal-oriented estimates for elliptic and parabolic problems that combines the dual-weighted residual method with equilibrated flux and potential reconstruction. These frameworks allow to analyze simultaneously different approximation schemes for the space discretization of the primal and the dual problems such as conforming or nonconforming finite element methods

State of the art domain decomposition algorithms for large-scale boundary value problems (with M≫1 degrees of freedom) suffer from bounded strong scalability because they involve the synchronisation and communication of workers inherent to iterative linear algebra. Here, we introduce PDDSparse, a different approach to scientific supercomputing which relies on a “Feynman-Kac formula for domain decomposition”

The parallel space-time adaptivity techniques for solving a cancer invasion model are investigated in the present work. Mathematically, the model comprises three coupled reaction-diffusion equations that characterize the cancer cell density evolution, the matrix-degrading enzymes and the extracellular matrix. The numerical realization demands fine resolutions both in spatial and temporal due to the

Adaptive Erlangization is a flexible observation/intervention strategy for system management based on randomized observation frequencies. The main contributions of this paper are the derivation and computation of some limit equation arising in the stochastic control under ambiguity along with its engineering application. In particular, with a focus on environmental management, we consider a novel long-term

The gas kinetic Bhatnagar-Gross-Krook (BGK) scheme proposed by Xu achieved great success and has been receiving a lot of attention during the past two decades. In this work, we proposed a new gas kinetic Bhatnagar-Gross-Krook (BGK) scheme for viscous flows. The proposed gas kinetic scheme is derived from the characteristic solution rather than the formal integral solution of the BGK model equation

Despite their well-established efficiency and accuracy, fractional-step schemes are not commonly used in finite volume methods. This article presents first-, second-, and third-order in-time fractional step schemes to solve incompressible convective time-dependent flows using collocated meshes. The fractional step methods are designed from a fully discrete problem and allow for optimal convergence

In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a fixed bulk mesh to the space-time surface. The structure of the method is such that it naturally fits to a level set representation of the evolving surface. The higher