A splitting positive definite mixed element method for second‐order hyperbolic equations

In this article, we establish a new mixed finite element procedure, in which the mixed element system is symmetric positive definite, to solve the second‐order hyperbolic equations. The convergence of the mixed element methods with continuous‐ and discrete‐time scheme is proved. And the corresponding error estimates are given. Finally some numerical results are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A reduced‐order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain

In this article, we primarily focuses to study the order‐reduction for the classical natural boundary element (NBE) method for the two‐dimensional (2D) hyperbolic equation in unbounded domain. To this end, we first build a semi‐discretized format about time for the hyperbolic equation and discuss the existence, stability, and convergence of the time semi‐discretized solutions. We then establish the classical fully discretized NBE format from the time semi‐discretized one and analyze the existence, stability, and convergence of the classical NBE solutions. Next, using proper orthogonal decomposition method, we build a reduced‐order extrapolated NBE (ROENBE) format containing very few unknowns but having adequately high accuracy, and we also discuss the existence, stability, and convergence of the ROENBE solutions. Finally, we use some numerical examples to show that the ROENBE method is far superior to the classical NBE one. It shows that the ROENBE method is reliable and effective for solving the 2D hyperbolic equation with the unbounded domain.     

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

We study a second order hyperbolic initial‐boundary value partial differential equation (PDE) with memory that results in an integro‐differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016     

Error estimates for mixed finite element approximations of the viscoelasticity wave equation

This paper studies mixed finite element approximations to the solution of the viscoelasticity wave equation. Two new transformations are introduced and a corresponding system of first‐order differential‐integral equations is derived. The semi‐discrete and full‐discrete mixed finite element methods are then proposed for the problem based on the Raviart–Thomas–Nedelec spaces. The optimal error estimates in L²‐norm are obtained for the semi‐discrete and full‐discrete mixed approximations of the general viscoelasticity wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

Full-Discrete Finite Element Method for Stochastic Hyperbolic Equation

This paper is concerned with the finite element method for the stochastic wave equation and the stochastic elastic equation driven by space-time white noise. For simplicity, we rewrite the two types of stochastic hyperbolic equations into a unified form. We convert the stochastic hyperbolic equation into a regularized equation by discretizing the white noise and then consider the full-discrete finite element method for the regularized equation. We derive the modeling error by using "Green's method" and the finite element approximation error by using the error estimates of the deterministic equation. Some numerical examples are presented to verify the theoretical results.     

A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation

An integro-differential equation of hyperbolic type, with mixed boundary conditions, is considered. A continuous space-time finite element method of degree one is formulated. A posteriori error representations based on space-time cells is presented such that it can be used for adaptive strategies based on dual weighted residual methods. A posteriori error estimates based on weighted global projections and local projections are also proved.     

Implicit–explicit multistep characteristic finite element methods for nonlinear convection–diffusion equations

Implicit–explicit multistep characteristic methods are given for convection‐dominated diffusion equations. Multistep difference along characteristics of the one‐order hyperbolic part of the equation is used for discretization in time, and finite element method is used to discrete the space variables. The resulting schemes are consistent, stable and very efficient. Optimal‐rate of convergence is proved. Also, a note is given for a paper published earlier© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A characteristics-mixed finite element method for Burgers’ equation

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers’ equation. This method is based upon a space-time variational form of Burgers’ equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates inL ²(Ω) are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.     

Splitting positive definite mixed element methods for pseudo‐hyperbolic equations

Splitting positive definite mixed finite element (SPDMFE) methods are discussed for a class of second‐order pseudo‐hyperbolic equations. Depending on the physical quantities of interest, two methods are proposed. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete schemes are proved. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 670–688, 2012     

Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type with weak dissipation

We consider a hyperbolic–parabolic singular perturbation problem for a quasilinear hyperbolic equation of Kirchhoff type with dissipation weak in time. The purpose of this paper is to give time‐decay convergence estimates of the difference between the solutions of the hyperbolic equation above and those of the corresponding parabolic equation, together with the unique existence of the global solutions of the hyperbolic equation above. Copyright © 2009 John Wiley & Sons, Ltd.     

Some non‐local problems for the parabolic‐hyperbolic type equation with complex spectral parameter

In the present paper the unique solvability of two non‐local problems for the mixed parabolic‐hyperbolic type equation with complex spectral parameter is proved. Sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is proved by the method of integral equations. In particular cases, eigenvalues and corresponding eigenfunctions of the studied problems are found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An optimal‐order error estimate for a Galerkin‐mixed finite‐element time‐stepping procedure for porous media flows

This article deals with the numerical approximation of miscible displacement problem of one incompressible fluid in a porous medium. The adopted formulation is based on the combined use of a mixed finite‐element scheme to treat pressure equation and of the finite‐element approach to treat concentration equation. Optimal‐order error estimates are obtained under some milder mesh‐parameter constraints. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 707–719, 2012     

An expanded mixed covolume method for sobolev equation with convection term on triangular grids

The expanded mixed covolume method for the two‐dimensional Sobolev equation with convection term is developed and studied. This method uses the lowest‐order Raviart‐Thomas mixed finite element space as the trial function space. By introducing a transfer operator γ_h which maps the trial function space into the test function space and combining expanded mixed finite element with mixed covolume method, the continuous‐in‐time, discrete‐in‐time expanded mixed covolume schemes are constructed, and optimal error estimates for these schemes are obtained. Numerical results are given to examine the validity and effectiveness of the proposed schemes.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media

A miscible displacement of one compressible fluid by another in a porous medium is governed by a nonlinear parabolic system. A new mixed finite element method, in which the mixed element system is symmetric positive definite and the flux equation is separated from pressure equation, is introduced to solve the pressure equation of parabolic type, and a standard Galerkin method is used to treat the convection‐diffusion equation of concentration of one of the fluids. The convergence of the approximate solution with an optimal accuracy in L²‐norm is proved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 229–249, 2001     

High order implicit collocation method for the solution of two‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving the two dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component. The resulted method is unconditionally stable and solves the two‐dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Sobolev-Volterra投影与积分微分方程有限元数值分析

本文提出一类称之为Sobolev-Volterra投影的有限元投影,研究了有关性质并将之应用于伪抛物型积分微分方程有限元方法、伪双曲型积分微分方程有限元方法以及三维伪双曲型积分微分方程交替方向有限元方法的数值分析.     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

The method of mixed volume element‐characteristic mixed volume element and its numerical analysis for three‐dimensional slightly compressible two‐phase displacement

Numerical simulation of oil‐water two‐phase displacement is a fundamental problem in energy mathematics. The mathematical model for the compressible case is defined by a nonlinear system of two partial differential equations: (1) a parabolic equation for pressure and (2) a convection‐diffusion equation for saturation. The pressure appears within the saturation equation, and the Darcy velocity controls the saturation. The flow equation is solved by the conservative mixed volume element method. The order of the accuracy is improved by the Darcy velocity. The conservative mixed volume element with characteristics is applied to compute the saturation, that is, the diffusion is discretized by the mixed volume element and convection is computed by the method of characteristics. The method of characteristics has strong computational stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. Small time truncation error and accuracy are obtained through this method. The mixed volume element simulates diffusion, saturation, and the adjoint vector function simultaneously. By using the theory and technique of a priori estimates of differential equations, convergence of the optimal second order in norm is obtained. Numerical examples are provided to show the effectiveness and viability of this method. This method provides a powerful tool for solving challenging benchmark problems.     

On the use of characteristic variables in viscoelastic flow problems

The system of partial differential equations governing the flow of an upper converted Maxwell fluid is known to be of mixed elliptic–hyperbolic type. The hyperbolic nature of the constitutive equation requires that, where appropriate, inflow conditions are prescribed in order to obtain a well-posed problem. Although there are three convective derivatives in the constitutive equation there are only two characteristic quantities whichare transported along the streamlines. These characteristicquantities are identified. A spectral element method is describedin which continuity of the characteristic variables is usedto couple the extra stress components between contiguous elements.The continuity of the characteristic variables is treated asa constraint on the constitutive equation. These conditionsdo not necessarily impose continuity on the extra-stress components.The velocity and pressure follow from the doubly constrainedweak formulation which enforces a divergence-free velocityfield and irrotational polymeric stress forces. This meansthat both the pressure and the extra-stress tensor are discontinuous.Numerical results are presented to demonstrate this procedure.The theory is applied to the upper convected Maxwell modelwith vanishing Reynolds number. No regularization techniques such as streamline upwind Petrov Galerkin (SUPG), elastic viscous split stress (EVSS) or explicitly elliptic momentum equation(EEME) are used.