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Max-Norm Error Estimates for Finite Element Methods for Nonlinear sobolev equations
CHOU, SO-HSIANG,LI, QIAN 한국산업정보응용수학회 2001 한국산업정보응용수학회 Vol.5 No.2
We consider the finite element method applied to nonlinear Sobolev equation with smooth data and demonstrate for arbitrary order (k≥2) finite element spaces the optimal rate of convergence in L_(∞) (W^(1,∞)(Ω)) and L_(∞) (L_(∞)(Ω)) (quasi-optimal for k = 1). In other words, the nonlinear Sobolev equation can be approximated equally well as its linear counterpart. Furthermore, we also obtain superconvergence results in L_(∞) (W^(1,∞)(Ω)) for the difference between the approximate solution and the generalized elliptic projection of the exact solution.
Comparing Two Approaches of Analyzing Mixed Finite Volume Methods
Chou, So-Hsiang,Tang, Shengrong 한국산업정보응용수학회 2001 한국산업정보응용수학회 Vol.5 No.1
Given the anisotropic Poisson equation -∇ · K∇p = f, one can convert it into a system of two first order PDEs: the Darcy law for the flux u = -K∇p and conservation of mass ∇· u = f. A very natural mixed finite volume method for this system is to seek the pressure in the nonconforming P1 space and the Darcy velocity in the lowest order Raviart-Thomas space. The equations for these variables are obtained by integrating the two first order systems over the triangular volumes. In this paper we show that such a method is really a standard finite element method with local recovery of the flux in disguise. As a consequence, we compare two approaches in analyzing finite volume methods (FVM) and shed light on the proper way of analyzing non co-volume type of FVM. Numerical results for Dirichlet and Neumann problems are included.