Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

The Yang-Mills flow in four dimensions

Global existence and uniqueness is established for the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy. Our results are analogous to those valid for the evolution of harmonic maps of Riemannian surfaces.The author gratefully acknowledges the hospitality and support of Princeton University, the Institute for Advanced Study, and the École Normale Supérieure Cachan.This article was processed by the author using the LaT_EX style file pljourl from Springer-Verlag.     

The singular dynamic method for constrained second order hyperbolic equations: Application to dynamic contact problems

The purpose of this paper is to present a new family of numerical methods for the approximation of second order hyperbolic partial differential equations submitted to a convex constraint on the solution. The main application is dynamic contact problems. The principle consists in the use of a singular mass matrix obtained by the mean of different discretizations of the solution and of its time derivative. We prove that the semi-discretized problem is well-posed and energy conserving. Numerical experiments show that this is a crucial property to build stable numerical schemes.     

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes

This work is concerned with the numerical capture of stiff viscous shock solutions of Navier-Stokes equations for complex compressible materials, in the regime of large Reynolds numbers. After [2] and [6], a relevant numerical capture is known to require the satisfaction of an extended set of non classical Rankine-Hugoniot conditions due to the non conservation form of the governing PDE model. Here, we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems. Non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate Riemann solutions. A relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method.     

Numerical approximation of the Cahn-Larché equation

Summary Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation - a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account. Here, we present a discretization based on finite elements and an implicit Euler scheme. We first show solvability and uniqueness of solutions. Based on an energy decay property we then prove convergence of the scheme. Finally we present numerical experiments showing the impact of elasticity on the morphology of the microstructure.Research supported by DFG Priority Program Analysis, Modeling and Simulation of Multiscale Problems under AR234/5-2 and GA695/2-2     

Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.     

A parallel four step domain decomposition scheme for coupled forward–backward stochastic differential equations

Motivated by the idea of imposing paralleling computing on solving stochastic differential equations (SDEs), we introduce a new domain decomposition scheme to solve forward–backward stochastic differential equations (FBSDEs) parallel. We reconstruct the four step scheme in Ma et al. (1994) [1] and then associate it with the idea of domain decomposition methods. We also introduce a new technique to prove the convergence of domain decomposition methods for systems of quasilinear parabolic equations and use it to prove the convergence of our scheme for the FBSDEs.     

Large-time geometric properties of solutions of the evolution p-Laplacian equation

We establish the behavior of the solutions of the degenerate parabolic equation

     

Operator-splitting methods in respect of eigenvalue problems for nonlinear equations and applications for Burgers equations

In this article we consider iterative operator-splitting methods for nonlinear differential equations with respect to their eigenvalues. The main focus of the proposed idea is the fixed-point iterative scheme that linearizes our underlying equations. On the basis of the approximated eigenvalues of such linearized systems we choose the order of the operators for our iterative splitting scheme. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator-splitting methods by providing improved results and convergence rates. We apply our results to deposition processes.     

Generalized stochastic evolution equations

An approach to generalized stochastic evolution equations is presented which is based on a generalized Ito formula. This allows the consideration of interesting examples which are stochastic generalizations of evolution equations of mixed type or second order in time hyperbolic equations. It includes more standard material involving a Gelfand triple of spaces as a special case. Several examples are given which illustrate the use of the abstract theory presented.     

NUMERICAL ANALYSIS OF A CONTACT PROBLEM IN RATE-TYPE VISCOPLASTICITY

In this paper, we consider numerical approximations of a contact problem in rate-type viscoplasticity. The contact conditions are described in term of a subdifferential and include as special cases some classical frictionless boundary conditions. The contact problem consists of an evolution equation coupled with a time-dependent variational inequality. Error estimates for both spatially semi-discrete and fully discrete solutions are derived and some convergence results are shown. Under appropriate regularity assumptions on the exact solution, error estimates are obtained.     

STABILITY ANALYSIS OF NONLINEAR GALERKIN ＆ GALERKIN METHODS FOR NONLINEAR EVOLUTION EQUATIONS

Abstract. Ogr object in this artlcle is to describe tbe Galerkln scheme and nonlin-eax Galerkin scheme for the approximation of nonlinear evolution equations, and tostudy the stability of these schemes. Spatial discretizatlon can be pedormed by eitherGalerkln spectral method or nonlinear Galerldn spectral method; time discretizatlort isdone hy Euler sin.heine wklch is explicit or implicit in the nonlinear terms. According tothe stability analysis of the above schemes, the stability of nonllneex Galerkln methodis better than that of Galexkln method.     

Large deviation principles for the stochastic quasi-geostrophic equations

In this paper we establish the large deviation principle for the stochastic quasi-geostrophic equation with small multiplicative noise in the subcritical case. The proof is mainly based on the weak convergence approach. Some analogous results are also obtained for the small time asymptotics of the stochastic quasi-geostrophic equation.     

On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations

In the paper, we extend Jörgens, Calabi, and Pogorelov's theorem on entire solutions of elliptic Monge-Ampère equations to parabolic equations associated with Gauss curvature flows. Our results include Gutiérrez and Huang's previous work as a special case. Besides, we also treat the isolated singularities for parabolic Monge-Ampère equations that was firstly studied by Jörgens for elliptic case in two dimensions.     

Solving parabolic and hyperbolic equations by the generalized finite difference method

Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points.     

An anisotropic area-preserving flow for convex plane curves

This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.     

Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities

The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term _∂tu${\partial }_{t}u$ is replace by _∂tb(u)${\partial }_{t}b\left(u\right)$, with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b$b$ is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitz-continuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b$b$ to be Lipschitz continuous.     

Laplace asymptotics for reaction-diffusion equations

Summary We obtain sharp (i.e. non logarithmic) asymptotics for the solution of non homogeneous Kolmogorov-Petrovski-Piskunov equation depending on a small parameter , for points ahead of the Freidlin-KPP front.