A discrete finite element galerkin method for a unidimensional single-phase Stefan problem

Based on Landau-type transformation, a Stefan problem with nonlinear free boundary condition is transformed into a system consisting of parabolic equation and the ordinary differential equations. Semidiscrete approximations are constructed. Optimal orders of convergence of semidiscrete approximation inL ₂,H ¹ andH ² normed spaces are derived.     

L ∞-convergence of collocation and galerkin approximations to linear two-point parabolic problems

Two semidiscrete collocation approximations using smooth cubic splines are developed as approximations to the solution of two-point linear parabolic boundary value problems.L _∞-convergence results are presented for these two approximations as well as the piecewise linear Galerkin approximation. Several computational examples are given to illustrate the convergence results and demonstrate the applicability of the method.     

Error estimates for a Galerkin method for a class of model equations for long waves

The Galerkin method, together with a second order time discretization, is applied to the periodic initial value problem for $$\frac{\partial }{{\partial t}}(u - (a(x)u_x )_x ) + (f(x,u))_x = 0$$ . Heref(x, ·) may be highly nonlinear, but a certain cancellation effect is assumed for∫f(x, u) _x u. Optimal order error estimates inL ₂,H ₁, andL _∞ are derived for a general class of piecewise polynomial spaces.     

Extrapolated Crank-Nicolson approximation for a linear Stefan problem with a forcing term

In this paper, we apply finite element Galerkin method to a single-phase linear Stefan problem with a forcing term. We apply the extrapolated Crank-Nicolson method to construct the fully discrete approximation and we derive optimal error estimates in the temporal direction inL ²,H ¹ spaces.     

Error estimates for mixed finite element approximations to a linear Stefan problem

We introduce a semidiscrete mixed finite element approximation for the single-phase linear Stefan problem and show the unique existence of the approximation. And the optimal rate of convergence inL ² andH ¹ norms are derived.     

Error analysis of a mixed finite element approximation of the semilinear Sobolev equations

Based on a mixed finite element method, we construct semidiscrete approximations of the solution u and the flux term ?u+?u _t of the semilinear Sobolev equations. The existence and uniqueness of the semidiscrete approximations are demonstrated and the error estimates of optimal rate in L ² normed space are derived. And also we construct the fully discrete approximations of u and ?u+?u _t and analyze the convergence of optimal rate in L ² normed space.     

Almost sure convergence of a Galerkin approximation for SPDEs of Zakai type driven by square integrable martingales

This work describes a Galerkin type method for stochastic partial differential equations of Zakai type driven by an infinite dimensional càdlàg square integrable martingale. Error estimates in the semidiscrete case, where discretization is only done in space, are derived in L^p and almost sure senses. Simulations confirm the theoretical results.     

A finite element method for the Sivashinsky equation

The Sivashinsky equation is a nonlinear evolutionary equation of fourth order in space. In this paper we have analyzed a semidiscrete finite element method and completely discrete scheme based on the backward Euler method and Crank–Nicolson–Galerkin scheme. A linearized backward Euler method have been developed and error bounds are derived for an L² projection.     

A finite element method for a single phase quasilinear free boundary problem in one space dimension

In this paper, we apply finite element Galerkin method to a singlephase quasilinear Stefan problem with a forcing term. To construct the fully discrete approximation we apply the extrapolated Crank-Nicolson method and we derive the optimal order of convergence 2 in the temporal direction inL ²,H ¹ normed spaces.     

The finite element method for a degenerate hyperbolic partial differential equation

Cahlon has recently considered finite difference methods for the degenerate Cauchy problem (tu _t)_t=Δu,u(x, 0)=u ₀(x) withu ₀ analytic. In this paper existence of a unique solution is shown for a more general class of initial data. A smoothing property of the solution operator is then exhibited. The usual semidiscrete finite element method is considered. The approximation is shown to be stable and superconvergent with orderO(h ^2μ?2) inl ₂ andl _∞, whereμ?1 is the degree of the polynomials used. OptimalL ² andL ^∞ estimates are also derived.     

A priori error estimates of finite volume element method for hyperbolic optimal control problems

In this paper,optimize-then-discretize,variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation.A semi-discrete optimal system is obtained.We prove the existence and uniqueness of the solution to the semidiscrete optimal system and obtain the optimal order error estimates in L ∞(J;L 2)-and L ∞(J;H 1)-norm.Numerical experiments are presented to test these theoretical results.     

Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

A second order finite difference-spectral method for space fractional diffusion equations

A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,respectively.Numerical experiments are carried out to demonstrate the theoretical analysis.     

Extension of isometries on the unit sphere of L p spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.     

On Kirchhoff's Model of Parabolic Type

In this article, the existence of a global strong solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, the existence of a global attractor is shown to hold for the problem when the non-homogeneous forcing function is either independent of time or in L^∞(L²). With the finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed, and it is also established that the semidiscrete system has a global discrete attractor. Optimal error estimates in L^∞(H¹) norm are derived which are valid uniformly in time. Further, based on a backward Euler method, a completely discrete scheme is analyzed and error estimates are derived. It is also further, observed that in cases where f = 0 or f = O(e^?γ₀t) with γ₀ > 0, the discrete solutions and error estimates decay exponentially in time. Finally, some numerical experiments are discussed which confirm our theoretical findings.     

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L₂ projection operator. Optimal strong convergence error estimates in the L₂ and norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case. AMS subject classification (2000) 65M, 60H15, 65C30, 65M65.Received April 2004. Revised September 2004. Communicated by Anders Szepessy.     

Galerkin methods for the time dependent incompressible elasticity equations

We consider semidiscrete and single step fully discrete approximations to the solutions of the time dependent incompressible elasticity equations. Two methods of dealing with the constraint are analysed: (a) a formulation involving a ‘discrete’ form of incompressibility condition and (b) a penalty formulation. H¹- and L²-error estimates are obtained.     

Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

Cesàro summability of one- and two-dimensional Walsh-Fourier series

ВВОДьтсьp-кВАжИлОкАл ьНыЕ ОпЕРАтОРы И ОДНО МЕРНыЕ ДИАДИЧЕскИЕ МАРтИНг АльНыЕ пРОстРАНстВА хАРДИH _p . ДОкАжАНО, ЧтО ЕслИ сУБлИНЕИНыИ ОпЕРАтО РT p-кВАжИлОкАлЕН И ОгРА НИЧЕН ИжL _∞ ВL _∞, тО ОН ьВльЕтсь тАкжЕ ОгРАН ИЧЕННыМ ИжH _p ВL _p , (0<p<1). В кАЧЕстВЕ пРИ лОжЕНИь ДОкАжАНО, ЧтО МАксИМАльНыИ ОпЕРАт ОР ОДНОгО ЧЕжАРОВскОгО пАРАМЕтРА И МОДИФИцИ РОВАННых ЧЕжАРОВскИх сРЕДНИх МАРтИНгАлА ьВльЕтсь ОгРАНИЧЕННыМ ИжH _p ВL _p И ИМЕЕт слАБыИ тИп (L ₁,L ₁). Мы ВВОДИМ ДВУМЕРНыИ ДИА ДИЧЕскИИ гИБРИД пРОс тРАНстВ хАРДИH ₁ И пОкАжыВАЕМ, Ч тО МАксИМАльНыИ ОпЕРАт ОР сРЕДНИх ЧЕжАРО ДВУ МЕРНОИ ФУНкцИИ ИМЕЕт слАБыИ тИп (H ₁ ^# ,L ₁). тАк Мы пОлУЧАЕМ, Ч тО ДВУпАРАМЕтРИЧЕск ИЕ сРЕДНИЕ ЧЕжАРО ФУНкц ИИf ?H ₁ ^# ?L logL схОДьтсь пОЧтИ ВсУДУ к ИсхОДНОИ ФУНк цИИ.     

A posteriori error estimates for the coupling equations of scalar conservation laws

In this paper we prove a posteriori L ₂(L ₂) and L _∞(H ^?1) residual based error estimates for a finite element method for the one-dimensional time dependent coupling equations of two scalar conservation laws. The underlying discretization scheme is Characteristic Galerkin method which is the particular variant of the Streamline diffusion finite element method for δ=0. Our estimate contains certain strong stability factors related to the solution of an associated linearized dual problem combined with the Galerkin orthogonality of the finite element method. The stability factor measures the stability properties of the linearized dual problem. We compute the stability factors for some examples by solving the dual problem numerically.