Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa- tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches....Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa- tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that （Vh（U -- Ihu）,VhVh）h may be estimated as order O（h2） when u E H3（Ω）, where Iuu denotes the bilinear interpolation of u, vh is a polynomial belongs to quasi-Wilson finite element space and △h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O（h2）/O（h3） in broken Hi-norm, which is one/two order higher than its interpolation error when u ε Ha（Ω）/H4 （（1）. Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O（h3）, two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.展开更多
In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different technique...In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product △↓（u - Ih^1u）, △↓vh） and the consistency error can be estimated as order O（h^2） in broken H^1 - norm/L^2 - norm when u ∈ H^3（Ω）/H^4（Ω）, where Ih^1u is the bilinear interpolation of u, Vh belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order O（h^2） for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order O（h^2 ＋ τ^2） is obtained for the rectangular partition when u ∈ H^4（Ω）, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.展开更多
基金Supported by National Natural Science Foundation of China (Grant Nos. 10971203 11101381+4 种基金 11271340)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20094101110006)Tianyuan Mathematics Foundation of the National Natural Science Foundation of China(Grant No. 11026154)the Natural Science Foundation of Henan Province (Grant Nos. 112300410026 122300410266)
文摘Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa- tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that （Vh（U -- Ihu）,VhVh）h may be estimated as order O（h2） when u E H3（Ω）, where Iuu denotes the bilinear interpolation of u, vh is a polynomial belongs to quasi-Wilson finite element space and △h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O（h2）/O（h3） in broken Hi-norm, which is one/two order higher than its interpolation error when u ε Ha（Ω）/H4 （（1）. Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O（h3）, two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.
文摘In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product △↓（u - Ih^1u）, △↓vh） and the consistency error can be estimated as order O（h^2） in broken H^1 - norm/L^2 - norm when u ∈ H^3（Ω）/H^4（Ω）, where Ih^1u is the bilinear interpolation of u, Vh belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order O（h^2） for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order O（h^2 ＋ τ^2） is obtained for the rectangular partition when u ∈ H^4（Ω）, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.
