Error estimates for discontinuous Galerkin method for nonlinear parabolic equations

We consider the nonlinear parabolic partial differential equations. We construct a discontinuous Galerkin approximation using a penalty term and obtain an optimal L^∞(L²) error estimate.     

具有散度形式的拟线性椭圆型微分方程的区域振动准则

对一类具有散度形式的拟线性椭圆型微分方程建立了若干新的振动准则,所得结果仅依赖于方程在外区域Ω(?)R~n的一个区域序列的信息而有别于已知的大多数结论.     

散度形式椭圆型方程弱下解的局部估计

运用De Giorgi迭代技巧,得到了一般散度形式的椭圆型偏微分方程弱下解的局部估计,推广了有关文献中的结论.     

A priori estimates for quasilinear degenerate parabolic equations

We prove some maximum and gradient estimates for classical solutions to a wide class of quasilinear degenerate parabolic equations, including first order ones. The proof is elementary and exploits the smallness of the domain in the time direction.

     

Regularity of solutions of linear and quasilinear equations of elliptic type in divergence form

Translated from Matematicheskie Zametki, Vol. 53, No. 1, pp. 68–82, January, 1993.     

Implicit-explicit multistep methods for quasilinear parabolic equations

Summary. Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation. Received March 10, 1997 / Revised version received March 2, 1998     

A new maximum principle of elliptic differential equations in divergence form

In this paper will be presented a new maximum principle of elliptic differential equations in divergence form which can be regarded as the counterpart of the Alexandroff-Bakelman-Pucci maximum principle of elliptic differential equations in nondivergence form.

     

Discontinuous mixed penalty-free Galerkin method for second-order quasilinear elliptic equations

Discrete schemes for finding an approximate solution of the Dirichlet problem for a second-order quasilinear elliptic equation in conservative form are investigated. The schemes are based on the discontinuous Galerkin method (DG schemes) in a mixed formulation and do not involve internal penalty parameters. Error estimates typical of DG schemes with internal penalty are obtained. A new result in the analysis of the schemes is that they are proved to satisfy the Ladyzhenskaya-Babuska-Brezzi condition (inf-sup) condition.     

Some recent methods for partial differential equations of divergence form

Some recent methods for solving second-order nonlinear partial differential equations of divergence form and related nonlinear problems are surveyed. These methods include entropy methods via the theory of divergence-measure fields for hyperbolic conservation laws, kinetic methods via kinetic formulations for degenerate parabolichyperbolic equations, and free-boundary methods via free-boundary iterations for multidimensional transonic shocks for nonlinear equation of mixed elliptic-hyperbolic type. Some recent trends in this direction are also discussed.Dedicated to IMPA on the occasion of its 50^th anniversary     

Partial Schauder estimates for second-order elliptic and parabolic equations

We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H?lder semi-norms not with respect to all, but only with respect to some of the independent variables.     

Some degenerate and quasilinear parabolic systems not in divergence form

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: u_t=u^p(Δu+av), v_t=v^q(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ₁², where λ₁ is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.     

Error estimates for a discontinuous Galerkin method for elliptic problems

In this paper, we consider an elliptic problem with the homogeneous Dirichlet boundary condition and introduce discontinuous Galerkin approximations of the problem. Optimal error estimates of discontinuous Galerkin approximations are obtained.     

Decay estimates for second-order quasilinear partial differential equations

This article is concerned with second-order quasilinear equations (not necessarily elliptic) in two independent variables of the form div[?(x, u, grad u)grad u] = 0. For Dirichlet (or Neumann) problems on a semi-infinite strip, with non-zero data on one end only, differential inequality techniques are employed to establish exponential decay estimates for quadratic (“energy”) integrals and to obtain pointwise estimates. The results have application to Saint-Venant principles for nonlinear elasticity and compressible fluid flows as well as to theorems of Phragmén-Lindelöf type.