Optimal Error Estimates Research Articles

Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue.

This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by using the Galerkin method combined with a compactness argument. A semi-discrete finite-element method and a fully discrete scheme using the Euler time-stepping have been formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented, along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed.

Optimal long time error estimates of a second-order decoupled finite element method for the Stokes–Darcy problem

In this paper, we propose a second-order decoupled finite element method based on lagging a part of the interfacial coupling terms for the time dependent Stokes–Darcy problem, which only need to solve two sub-physical problems sequentially. Under a modest time step restriction Δt≤C(physical parameters), the optimal long time error estimates are obtained both in the L2 norm and in the H1 norm. Numerical results are provided to illustrate the convergence rate O(Δt2) of the temporal approximation.

Unconditional stability and error estimates of the modified characteristics FEMs for the Micropolar Navier–Stokes Equations

In this paper, the unconditional stability and optimal error estimate of the velocity, pressure and angular velocity for the modified characteristics FEMs of the unsteady Micropolar Naiver–Stokes Equations (MNSE) are presented. In this method, the nonlinear equation is linearized by the characteristic finite element method for dealing with the time derivative term and the convection term. Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the error between the numerical solution and the solution of the time-discrete system is τ-independent, where τ denotes the time stepsize. The stability results and optimal error estimates in L2 norm and H1 norm will be given. Finally, some numerical results will be provided to confirm our theoretical analysis.

A Crank–Nicolson leap-frog scheme for the unsteady incompressible magnetohydrodynamics equations

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style’s scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm.

Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

Abstract The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

Discontinuous Galerkin method for the coupled dual-porosity-Brinkman model

In this paper we use the weighted discontinuous Galerkin finite element method to solve the dual-porosity-Brinkman coupling model. The dual-porosity model describes fluid flow in a dual-porosity medium composed of matrix and microfractures, while the Brinkman equation describes fluid flow in hydraulic fractures or conduits containing “debris filling” or “proppant filling”. The two models are coupled by the four physically effective interface conditions, which are constructed based on the fundamental properties of the dual-porosity model and the Brinkman-Darcy model. We provide a weighted discontinuous discrete scheme for the model and derive the optimal error estimates. Through numerical experiments, we verify the properties of the model and the advantages of the numerical method, such as the optimal convergence rate of the numerical solution and the simulation of the flow characteristics around hydraulic fractures and conduits.

An exact in time Fourier pseudospectral method with multiple conservation laws for three-dimensional Maxwell’s equations

Maxwell’s equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell’s equations. This is accomplished by combining Fourier pseudospectral methods in space and exact formulation in time. Fast computation is efficiently implemented in the scheme by using the matrix diagonalisation method and fast Fourier transform algorithm which are well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and exact in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

AbstractThis article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis.

Analysis of the boundary conditions for the ultraweak-local discontinuous Galerkin method of time-dependent linear fourth-order problems

In this paper, we study the ultraweak-local discontinuous Galerkin (UWLDG) method for time-dependent linear fourth-order problems with four types of boundary conditions. In one dimension and two dimensions, stability and optimal error estimates of order k + 1 k+1 are derived for the UWLDG scheme with polynomials of degree at most k k ( k ≥ 1 k\ge 1 ) for solving initial-boundary value problems. The main difficulties are the design of suitable penalty terms at the boundary for numerical fluxes and the construction of projections. More precisely, in two dimensions with the Dirichlet boundary condition, an elaborate projection of the exact boundary condition is proposed as the boundary flux, which, in combination with some proper penalty terms, leads to the stability and optimal error estimates. For other three types of boundary conditions, optimal error estimates can also be proved for fluxes without any penalty terms when special projections are designed to match different boundary conditions. Numerical experiments are presented to confirm the sharpness of theoretical results.

Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.

A space–time DG method for the Schrödinger equation with variable potential

We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.

A unified immersed finite element error analysis for one-dimensional interface problems

It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

On the necessity of the inf-sup condition for a mixed finite element formulation

Abstract We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $\textbf{H}(\operatorname{\textrm{div}};\varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.

Convergence analysis of virtual element method for the electric interface model on polygonal meshes with small edges

A conforming virtual element method is studied for approximating the solution of a pulsed electric interface model on polygonal meshes with small edges or faces, which traditional virtual and finite element methods cannot easily handle. One of the significant advantages of this virtual element method is to generate interface-conforming meshes efficiently exploiting polygonal meshes and hanging nodes. The virtual element method employs specific projection operators to establish optimal error estimates with reasonable assumptions regarding the solution's regularity. The voltage potential of the pulsed electric model through the physical media is approximated by using a fully discrete virtual element approach based on Crank-Nicolson temporal discretization. Numerical examples are used to illustrate the expected order of convergence. The efficiency and robustness of the proposed method are displayed on interface-independent/dependent background-fitted meshes with small edges and large magnitudes of discontinuities across the interface by these experiments.

A discontinuous Galerkin finite element method for the Oldroyd model of order one

In this work, we analyze a discontinuous Galerkin finite element method for the equations of motion that arise in the 2D Oldroyd model of order one. We investigate the existence and uniqueness of semidiscrete discontinuous solutions, as well as the consistency of the scheme. We derive new a priori and regularity results for the discrete solution and establish optimal error estimates in ‐norm in time and energy norm in space for the velocity and ‐norm in both time and space for the pressure. Uniform estimates are derived for sufficiently small data. We next apply the backward Euler method to the semidiscrete formulation and establish optimal fully discrete error estimates. At the end, we conduct numerical experiments to support our theoretical results and analyze the findings.

An upwind-mixed volume element method on changing meshes for compressible miscible displacement problem

Numerical simulation of oil-water miscible displacement is discussed in this paper, and a compressible problem of energy mathematics is solved potentially. The mathematical model defined by a nonlinear system includes mainly two partial differential equations (PDEs): a parabolic equation for the pressure and a convection-diffusion equation for the saturation. The pressure is obtained by a conservative mixed finite volume element method (MFVE). The computational accuracy is improved for Darcy velocity. A conservative upwind mixed finite volume element method (UMFVE) is applied to compute the saturation on changing meshes. The diffusion is discretized by a mixed finite volume element method, and the convection is computed by upwind differences. The upwind method can solve convection-dominated diffusion equations accurately and avoids numerical dispersion and nonphysical oscillation. The saturation and the adjoint vector function are obtained simultaneously. An optimal order error estimates is obtained. Finally, numerical examples are provided to show the accuracy, efficiency and possible applications.

Low order nonconforming finite element methods for nuclear reactor model

A uniform framework of the linearized fully decoupled fully discrete schemes is developed and investigated with low order nonconforming finite element methods (FEMs) for nuclear reactor model. On the one hand, a general scheme called Scheme I is constructed. Then, the modified Ritz projection and mathematical induction are used to obtain its unconditional optimal error estimate in the L2-norm and superclose estimates in the broken H1-norm, thereby correcting the mistake in previous literature and eliminating the need for the so-called popular time-space splitting method. On the other hand, a new positivity-preserving scheme named Scheme II, which combines the traditional FEMs with cut-off post-processing method, is designed to avoid the non-physical oscillation in numerical calculations. Based on the results of Scheme I, an optimal error estimate in the L2-norm for Scheme II is also derived. Finally, taking the nonconforming EQ1rot element as an example, some numerical experiments are provided to demonstrate the theoretical results. It is worth noting that the analysis presented herein is also suitable for simulating nuclear reactor model in the narrow channel with anisotropic meshes.

Optimal Error Estimates of the Penalty Finite Element Method for the Unsteady Navier–Stokes Equations with Nonsmooth Initial Data

Optimal Error Estimates of the Penalty Finite Element Method for the Unsteady Navier–Stokes Equations with Nonsmooth Initial Data

A space-time Petrov-Galerkin method for the two-dimensional regularized long-wave equation

In this work, a space-time Petrov-Galerkin (STPG) method is used to numerically analyze the two-dimensional regularized long-wave (RLW) equation. The STPG method is a nonstandard finite element method, that is, both of the spatial and temporal variables of this method are discretized by finite element method. Therefore, it can display the superiority of the finite element method in both spatial and temporal directions. In particular, it is easy to obtain space-time high accuracy and is unconditionally stable, thus this method is very appropriate for solving the nonlinear convection-diffusion equations. We prove the existence and uniqueness of the approximate solution and derive an optimal order error estimate in the maximum norm without requiring any compatibility between the space and time mesh size. In addition, some numerical experiments are presented to validate the accuracy and efficiency of the proposed method. Also, we provide several numerical experiments to confirm the conservation properties of discrete mass, energy and momentum for both the single and double-soliton waves.