In this paper, we present a rigorous optimal error analysis for a semi-discrete finite element method of the coupled Cahn-Hilliard-Navier-Stokes-Darcy model. The optimal convergence order for the phase variable in the L infinity(0, T; H1) norm and the velocity in the L infinity(0, T; L2) norm are proved with two key techniques. One key technique is establishing the discrete L2(0, T; H2) boolean AND L2(0, T; L infinity) boundedness of the numerical solution for the phase variable to handle the nonlinear regional coupling terms. Another key technique adopts the inverse inequality and Ritz projection to bound the L infinity norm of the error of the phase variable in the nonlinear terms that depend on the viscosity and the mobility. This is the first convergence analysis for the sophisticated Cahn-Hilliard-Navier-Stokes-Darcy system. Numerical experiments are also presented to verify the theory and validate the spatial semi-discretization finite element scheme.

This article establishes a phase field model for governing the two-phase incompressible MHD flows with different densities, electric conductivities, and viscosities. In addition to the coupling between the Cahn-Hilliard phase field equations and the single-phase MHD equations together with the varying parameters, it is physically faithful and mathematically rigorous for the modeling to incorporate a relative flux term, which is related to the diffusion of the components, into the coupled system, inspired by Abels et al. and Shen and Yang. We present a linear fully discrete numerical scheme for this complex multi-physics system, which leverages the artificial compressibility method, an explicit-implicit treatment of nonlinear terms, and the addition of several key stabilization terms. This scheme is proven uniquely solvable per time step and unconditionally stable. And it is free from any artificial pressure boundary conditions. Accurate capture of the diffuse interface is achieved through an adaptive mesh strategy for local refinement of the interfacial region. To display the features and applicability of the presented model and scheme, we conduct a suite of numerical simulations, which include an accuracy test, the spinodal decomposition, one or three bubbles rising in magnetic fluid with large density ratios, the Rayleigh-Taylor instability, and the interface pinchoff.

This paper carries out the optimal L2 error estimation for a fully discrete method, which was proposed in [J. Sci. Comput., 95(1):16, 2023] for the Cahn-Hilliard-MHD model with constant coefficients. Compared with the original scheme, we adopt weaker regularity requirement for the finite element space of the discrete phase field. To obtain the optimal convergence order, H-1-superconvergence error estimates of Ritz projection and Ritz quasi-projection are employed. The H1 semi-norm and L2 norm terms of the phase-field system, arising from the H-1-H1 norm pair, are estimated by two sets of test functions. Furthermore, the first-order differential operator induces a loss of convergence rate in the projection error terms. Hence Ritz quasi-projection and Stokes quasi-projection, incorporating the above terms, are adopted from [SIAM J. Numer. Anal.,61(3):1218-1245, 2023]. Additionally, operator transfer based on Green's formula provides an alternative strategy. We also carry out numerical examples to validate the numerical scheme.

In this article, we analyze a semi-implicit noniterative decoupling method for the unsteady Navier-Stokes-Darcy system, based on multi-step backward differentiation schemes for the time discretization and finite elements for the spatial discretization. The results in the previous time steps are utilized to directly predict the interface information for decomposing the Navier-Stokes and Darcy sub-domains without any iteration at each time step. A semi-implicit scheme is proposed to linearize the nonlinear convection. In order to analyze the convergence of the finite-element solution of the proposed method, we rigorously prove the error estimate in the $L<^>{2}$ norm for the joint Stokes-Darcy Ritz-projection, notably without relying on the $H<^>{2}$ regularity assumption of the elliptic problem. The general $k$-step backward differentiation formulae ($1\le k\le 5$) are analyzed within a general framework utilizing multiplier techniques, and mathematical induction is employed to address the terms arising from nonlinear advection in the error estimate. Numerical examples are provided to verify the theoretical conclusions and illustrate the proposed method.

Stabilizer-free P_k virtual elements are constructed on polygonal and polyhedral meshes. Here the interpolating space is the space of continuous P_k polynomials on a triangular-subdivision of each polygon, or a tetrahedral-subdivision of each polyhedron. With such an accurate and proper interpolation, the stabilizer of the virtual elements is eliminated while the system is kept positive-definite. We show that the stabilizer-free virtual elements converge at the optimal order in 2D and 3D. Numerical examples are computed, validating the theory.

This paper proposes and analyzes a fully discrete semi-implicit unconditionally energy stable numerical scheme to solve the Cahn-Hilliard Magnetohydrodynamics (Cahn-Hilliard-MHD) model with variable density. The unconditional energy stability and optimal L2 error estimates are established for the fully discrete scheme. Major challenges in error estimation arise from the variable density, the strong nonlinearities, and the multi-physics coupling of the model. Under the mathematical induction framework, the Ritz quasi-projection and the Stokes quasi-projection, proposed in [SIAM J. Numer. Anal., 61(3):1218-1245, 2023], are utilized to avoid the gradient terms of the projection errors. The H-1 superconvergence error estimates of Ritz projection and Ritz quasi-projection reduce the regularity requirement for the finite element space. Two different sets of test functions are also selected to avoid extra estimations of the L2 norm terms. The summation by parts transfers the backward difference quotient operator from the test function to the trial function, avoiding the need for extra estimation of the test function. A numerical experiment is provided to verify the theoretical results.

This paper proposes, analyzes, and numerically solves an enhanced model for the two-phase ferro-fluid flows with different viscosities and densities. In addition to the variable parameters for the density and viscosity and a reformulation of the magnetostatic equation, a relative flux term related to the diffusion of the components is also incorporated into the model to ensure a physically consistent and mathematically rigorous system, inspired by Abels et al. (2012) and Shen and Yang (2015). The introduction of this relative flux term plays a pivotal role in the analysis of the energy law and the construction of the numerical scheme. To construct an efficient fully discrete finite element numerical scheme, we introduce an intermediate magnetization for the magnetization equation, add several key stabilization terms due to the explicit treatment of the nonlinear and coupling terms, apply the artificial compressibility approach for the Navier–Stokes equations, and utilize the invariant energy quadratization approach for the Cahn-Hilliard equations. It is proven to be uniquely solvable per time step and unconditionally stable. Meanwhile, the proposed scheme requires no artificial boundary conditions on the pressure. In addition, we employ an adaptive mesh strategy to better capture the diffuse interface in practical numerical simulations. We present several 2D and 3D numerical examples to demonstrate and validate the presented two-phase model and numerical scheme, including the accuracy test, the Spinodal decomposition, the deformation and migration of a ferrofluid droplet in an irregular domain, as well as the Rosensweig instability with large density ratios under the uniformly or nonuniformly applied magnetic field.

In this article, we propose and analyze a stabilized weighted interior penalty method for solving the thermal convection problems in heterogeneous porous media. We first transform the thermal convection model into the pressure primal form with homogeneous Neumann boundary condition, and develop a symmetric weighted interior penalty method to handle the discontinuous Darcy number and automatically adjust the penalty coefficient corresponding to the varying permeability. Then we recover the velocity by a stabilized method and incorporate it into the energy equation to obtain the temperature. The stability and convergence rates of the numerical solutions are rigorously proved, and verified by numerical tests. The numerical experiments also include the testing benchmarks for cavity flow problems featuring discontinuous heterogeneous Darcy numbers, to further validate the effectiveness of the proposed scheme.

A fully discrete low-regularity integrator with high-frequency recovery techniques is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method can capture the discontinuities of the solutions correctly without spurious oscillations and can approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that $(u,\partial_{t}u)\in C([0,T];H^{\gamma}\times H^{\gamma-1})$ for $\gamma\in(0,1]$. For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition $\tau \sim N^{-1}$, where $\tau$ and $N$ denote the time step size and the number of Fourier terms in the space discretization, respectively. Extensive numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method.

When solving the Poisson equation on honeycomb hexagonal grids, we show that the $P_1$ virtual element is three-order superconvergent in $H^1$-norm, and two-order superconvergent in $L^2$ and $L^\infty$ norms. We define a local post-process which lifts the superconvergent $P_1$ solution to a $P_3$ solution of the optimal-order approximation. The theory is confirmed by a numerical test.

A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn–Hilliard–Navier–Stokes–Darcy model in the coupled free fluid region and porous medium region. By introducing two auxiliary energy variables, we derive the equivalent system that is consistent with the original system. The energy dissipation law of the proposed equivalent model is proven. To lay a solid foundation, we first present a coupled linearized time-stepping method for the reformulated system, and prove its unconditionally energy stability. In order to further improve the computational efficiency, special treatment for the interface conditions and the artificial compression approach are utilized to decouple the two subdomains and the Navier–Stokes equation. Therefore, with the discretization techniques of two existing auxiliary variable approaches, a fully decoupled and linearized numerical scheme can be developed, under the framework of a semi-implicit semi-explicit scheme for temporal discretization and Galerkin finite element method for spatial discretization. The grad-div stabilization is also employed to further improve the stability of auxiliary variable algorithm. The full discretization obeys the desired energy dissipation law without any temporal restriction. Moreover, the implementation process is discussed, including the adaptive mesh strategy to accurately capture the diffuse interface. Ample numerical experiments are performed to validate the typical features of developed numerical schemes, such as the accuracy, energy stability without restriction for time step size, and adaptive mesh refinement in space. Furthermore, we apply the proposed numerical method to simulate the shape relaxation and the Buoyancy-driven flows, which demonstrate the applicability of the proposed method.

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.

This article is concerned with the question of whether it is possible to construct a time discretization for the one-dimensional cubic nonlinear Schrodinger equation with second-order convergence for initial data with regularity strictly below H-2. We address this question with a positive answer by constructing a new second-order low-regularity integrator for the one-dimensional cubic nonlinear Schrodinger equation. The proposed method can have second-order convergence in L(2)for initial data in H-3/(5) , and first-order convergence up to a logarithmic factor for initial data in H-1. This significantly relaxes the regularity requirement for second-order approximations to the cubic nonlinear Schrodinger equation, while retaining the by far best convergence order for initial data in H-1. Numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed method in approximating nonsmooth solutions of the nonlinear Schrodinger equation. The numerical results show that, among the many time discretizations, the proposed method is the only one that has second-order convergence in L-2 for initial data strictly below H-2.

A new spectral method is constructed for the linear and semilinear subdiffusion equations with possibly discontinuous rough initial data. The new method effectively combines several computational techniques, including the contour integral representation of the solutions, the quadrature approximation of contour integrals, the exponential integrator using the de la Valle'\e Poussin means of the source function, and a decomposition of the time interval geometrically refined towards the singularity of the solution and the source function. Rigorous error analysis shows that the proposed method has spectral convergence for the linear and semilinear subdiffusion equations with bounded measurable initial data and possibly singular source functions under the natural regularity of the solutions.

Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$ , and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$ -elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments.

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.

This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise H2 regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual L2 norm. A numerical example is conducted to validate the theoretical conclusions.

The paper is concerned with the multiscale analysis of the scattering problem for three-dimensional time-dependent Maxwell’s equations in heterogeneous materials. Firstly, an exact transparent boundary condition is developed to reduce the scattering problem into an initial–boundary value problem in heterogeneous materials. Secondly, the multiscale asymptotic expansions of the solution for the reduced problem and an explicit convergence rate for the approximate solutions are presented. Finally, a multiscale Crank–Nicolson mixed finite element method is proposed where the first-order approximation of the Silver–Müller radiation condition is utilized to truncate infinite domain problems. Numerical experiments are then carried out to validate the theoretical results.

In this paper, we consider the Galerkin finite element methods for the Maxwell- Klein-Gordon system in the Coulomb gauge. We propose a semidiscrete finite element method for the system with the mixed finite element approximation of the vector potential. Energy conservation and error estimates are established for this scheme. A novel energy conserving time integration scheme is presented for solving the semidiscrete system. The existence and uniqueness of solutions to the fully discrete system are proved under some assumptions. Numerical experiments are carried out to support our theoretical analysis.

In this paper, we are concerned with the system of Euler equations with time-dependent damping like $-\frac{\mu}{(1+t)^\lambda}u$ for physical parameters $\lambda\ge 0$ and $\mu>0$. It is well known that, when $0\le \lambda<1$, the time-asymptotic-degenerate damping plays the key role which makes the damped Euler system behave like time-degenerate diffusion equations, while, when $\lambda>1$, the damping effect becomes really weak and can be neglected, which makes the dynamic system essentially behave like a hyperbolic system, and the singularity of solutions like shock waves will form. However, in the critical case $\lambda=1$, when $0<\mu\le 2$, the solutions of the system will blow up, but when $\mu>2$, the system is expected to possess global solutions. Here, we are particularly interested in the asymptotic behavior of the solutions in the critical case. By a heuristical analysis (variable scaling technique), we realize that, in this critical case, the hyperbolicity and the damping effect both play crucial roles and cannot be neglected. We first artfully construct the asymptotic profile, a special linear wave equation with time-dependent damping, which is totally different from the case of $0\leq\lambda<1,\mu>0$, whose profile is a self-similar solution to the corresponding parabolic equation. Then we rigorously prove that the solutions time-asymptotically converge to the solutions of linear wave equations with critical time-dependent damping. The convergence rates shown are optimal, by comparing with the linearized equations. The proof is based on the technical time-weighted energy method, where the time-weight is dependent on the parameter $\mu$.