Optimal Order Error Estimates Research Articles

Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation

In this article we adopt the saddle-point theoretical framework to analyze the conservative fractional diffusion equations. By introducing a fractional-order flux as auxiliary variable, we establish the well-posedness of the saddle-point variational formulation. Based on the formulation we propose a locally-conservative expanded mixed finite element procedure to approximate the unknown, its derivative and the fractional flux directly, and prove the existence and uniqueness of the mixed finite element solution. By proving that the negative fractional derivative spaces are equivalent to the negative fractional-order Sobolev spaces we derive the approximation capability of the standard projection operator in fractional Sobolev spaces for non-smooth functions. This leads an optimal-order error estimates in terms of the right-hand side both for sufficiently smooth solution and non-smooth solution. Numerical experiments are included to confirm our theoretical findings.

Symmetric finite volume element approximations of second order linear hyperbolic integro-differential equations

In this paper, based on barycenter dual meshes, we develop one semi-discrete and two full discrete symmetric finite volume element schemes for second order linear hyperbolic integro-differential equations. The optimal order error estimates in L2 and H1-norms are derived for the semi-discrete scheme. Numerical experiments confirm the performance of the symmetric schemes, and further show that the L2-norm convergence rate of the full discrete backward Euler and Crank–Nicolson schemes to be O(h2+τ) and O(h2+τ2), respectively.

Theory and application of fractional step characteristic finite difference method in numerical simulation of second order enhanced oil production

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results are quite interesting and satisfactory.

Stabilized finite element methods for a blood flow model of arteriosclerosis

In this article, a blood flow model of arteriosclerosis, which is governed by the incompressible Navier–Stokes equations with nonlinear slip boundary conditions, is constructed and analyzed. By means of suitable numerical integration approximation for the nonlinear boundary term in this model, a discrete variational inequality for the model based on stabilized finite elements is proposed. Optimal order error estimates are obtained. Finally, numerical examples are shown to demonstrate the validity of the theoretical analysis and the efficiency of the presented methods. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2063–2079, 2015

A trace finite element method for a class of coupled bulk-interface transport problems

In this paper we study a system of advection-diffusion equations in a bulk domain coupled to an advection-diffusion equation on an embedded surface. Such systems of coupled partial differential equations arise in, for example, the modeling of transport and diffusion of surfactants in two-phase flows. The model considered here accounts for adsorption-desorption of the surfactants at a sharp interface between two fluids and their transport and diffusion in both fluid phases and along the interface. The paper gives a well-posedness analysis for the system of bulk-surface equations and introduces a finite element method for its numerical solution. The finite element method is unfitted, i.e. , the mesh is not aligned to the interface. The method is based on taking traces of a standard finite element space both on the bulk domains and the embedded surface. The numerical approach allows an implicit definition of the surface as the zero level of a level-set function. Optimal order error estimates are proved for the finite element method both in the bulk-surface energy norm and the L 2 -norm. The analysis is not restricted to linear finite elements and a piecewise planar reconstruction of the surface, but also covers the discretization with higher order elements and a higher order surface reconstruction.

Symmetric finite volume method for second order variable coefficient hyperbolic equations

In this paper, we consider one semi-discrete and two full discrete symmetric finite volume schemes for a class of second order variable coefficient hyperbolic equations based on a linear finite element space. The optimal order error estimates in L2, H1 norms are derived for the semi-discrete and full discrete schemes. Finally, some numerical experiments are presented to confirm the performance of the symmetric schemes.

P 1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order P1-nonconforming triangular finite element method (FEM) for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.

Domain Decomposition Modified with Characteristic Finite Element Method for Numerical Simulation of Semiconductor Transient Problem of Heat Conduction

A characteristic finite element algorithm based on domain decomposition is structured in this paper to approximate numerically multi-dimensional semiconductor transient problems of heat conduction. Finite element approximation is presented for the electric field potential equation, and a domain decomposition discretization with characteristic finite element is put forward for the electron concentration equation, hole concentration equation and heat conductor equation. An optimal order error estimate in L2 norm is derived for the coupled system by using some techniques such as variation, domain decomposition, the method of characteristics, the principle of energy, negative norm estimates, induction hypothesis, prior estimates theory and other techniques of partial differential equations. Finally, experimental data consistent with theoretical convergence rate are shown. This type of numerical method is of high computational efficiency and can successfully solve this international problem.

A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem

In this paper, a stabilized mixed finite element method for a coupled steady Stokes–Darcy problem is proposed and investigated. This method is based on two local Gauss integrals for the Stokes equations. Its originality is to use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the coupled Stokes–Darcy problem by using the lowest equal-order finite element triples. This new method has several attractive computational features: parameter free, flexible, and altering the difficulties inherited in the original equations. Stability and error estimates of optimal order are obtained by using the lowest equal-order finite element triples (P1−P1−P1) and (Q1−Q1−Q1) for approximations of the velocity, pressure, and hydraulic head. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the coupled problem with the Beavers–Joseph–Saffman–Jones and Beavers–Joseph interface conditions.

Unified analysis of higher-order finite volume methods for parabolic problems on quadrilateral meshes

In this paper, a unified analysis for higher-order finite volume methods for parabolic problems on quadrilateral meshes is presented. By studying the quasi-symmetry of the finite volume bilinear form, optimal-order error estimates in the |$ L^\infty (H^1)$|- and |$L^\infty (L^2)$|-norms are derived. The theoretical estimates are validated by numerical experiments.

Analysis of a contact problem with normal damped response and unilateral constraint

We consider a mathematical model which describes the equilibrium of a viscoelastic body in frictional contact with an obstacle. The contact is modeled with normal damped response and unilateral constraint for the velocity field, associated to a version of Coulomb's law of dry friction. We present a weak formulation of the problem, then we state and prove an existence and uniqueness result of the solution. The proof is based on arguments of history-dependent quasivariational inequalities. We also study the dependence of the solution with respect to the data and prove a convergence result. Further, we introduce a fully discrete scheme to solve the problem numerically. Under certain solution regularity assumptions, we derive an optimal order error estimate of the discretization. Finally, we provide numerical simulations which illustrate the behavior of the solution with respect to the frictional contact conditions and validate the theoretical convergence results.

Anisotropic Crouzeix-Raviart Type Nonconforming Finite Element Methods to Variational Inequality Problem with Displacement Obstacle

In this paper, anisotropic Crouzeix-Raviart type nonconforming finite element methods are considered for solving the second order variational inequality with displacement obstacle. The convergence analysis is presented and the optimal order error estimates are obtained under the hypothesis of the finite length of the free boundary. Numerical results are provided to illustrate the correctness of theoretical analysis.

A Modified Weak Galerkin Finite Element Method for Sobolev Equation

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete H 1 and L 2 norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.

Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

In this paper we analyze a low-order finite element method for approximating the vibration frequencies and modes of a non-homogeneous Timoshenko beam. We consider a formulation in which the bending moment is introduced as an additional unknown. Optimal order error estimates are proved for displacements, rotations, shear stress and bending moment of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are independent of the beam thickness, which leads to the conclusion that the method is locking free. For its implementation, displacements and rotations can be eliminated leading to a well posed generalized matrix eigenvalue problem for which the computer cost of its solution is similar to that of other classical formulations. We report numerical experiments which allow us to assess the performance of the method.

A weak Galerkin finite element method for the stokes equations

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree $k\ge 1$ for the velocity and polynomials of degree $k-1$ for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree $k-1$ on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.

A hybridized weak Galerkin finite element scheme for the Stokes equations

In this paper a hybridized weak Galerkin (HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced. The WG method uses weak functions and their weak derivatives which are defined as distributions. Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution. With this new feature, HWG method can be used to deal with jumps of the functions and their flux easily. Optimal order error estimate are established for the corresponding HWG finite element approximations for both {\color{black}primal variables} and the Lagrange multiplier. A Schur complement formulation of the HWG method is derived for implementation purpose. The validity of the theoretical results is demonstrated in numerical tests.

A stable nonconforming mixed finite element scheme for elliptic optimal control problems

A new stable nonconforming mixed finite element (MFE) scheme is proposed for the constrained optimal control problem (OCP) governed by elliptic equations. The nonconforming FEs are employed to approximate the co-state, the piecewise constant element for the state and control, respectively. The corresponding optimal order error estimates are derived by full use of the distinguish properties of the nonconforming element, i.e., (a) the divergence of the function in the FE space is a constant; (b) the consistency error is one order higher than its interpolation error in the broken energy norm. Finally, some numerical results are carried out to verify the theoretical analysis.

Theory and Application of Characteristic Finite Difference Fractional Step Method of Capillary Force Enhanced Oil Production

A kind of second-order implicit characteristic fractional steps finite difference method is presented in this paper for the numerical simulation coupled system of enhanced (chemical) oil production on consideration capillary force in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in $l^2$ norm is derived. This method has been applied successfully the numerical simulation of enhanced oil production in actual oilfields, and the simulation results are quite interesting and satisfactory.

A virtual element method for the Steklov eigenvalue problem

The aim of this paper is to develop a virtual element method for the two-dimensional Steklov eigenvalue problem. We propose a discretization by means of the virtual elements presented in [L. Beirão da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199–214]. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. We also prove higher-order error estimates for the computation of the eigensolutions on the boundary, which in some Steklov problems (computing sloshing modes, for instance) provides the quantity of main interest (the free surface of the liquid). Finally, we report some numerical tests supporting the theoretical results.

A finite element method for the buckling problem of simply supported Kirchhoff plates

The aim of this paper is to develop a finite element method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. We introduce an auxiliary variable w:=Δu (with u representing the displacement of the plate) to write a variational formulation of the spectral problem. We propose a conforming discretization of the problem, where the unknowns are approximated by piecewise linear and continuous finite elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments supporting our theoretical results.