Finite element approximation of some degenerate monotone quasilinear elliptic systems
Finite element approximation of some degenerate monotone quasilinear elliptic systems
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DOI：
10.1137/0733006
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发表时间：
1996-02
影响因子：
2.9
通讯作者：
Wenbin Liu;J. Barrett
中科院分区：
文献类型：
--
作者：
Wenbin Liu;J. Barrett
In this paper we examine the continuous piecewise linear finite element approximation of the following system: given ${\bf f} \equiv (f_j )$ and ${\bf g} \equiv (g_j )$, find ${\bf u} \equiv (u_j )$($(j = 1 \to r$ with $r = 1$ or 2) such that $ - \nabla \cdot (K(z,\nabla {\bf u}(z))\nabla {\bf u}(z)) = {\bf f}(z),\quad z \in \Omega \subset R^2 ,\quad {\bf u} |_{\partial \Omega } = {\bf g} |_{\partial \Omega } ,$ where $(\nabla {\bf u}) \equiv {{\partial u_j } / {\partial z_i }}\, 1 \leq i \leq 2,\, 1 \leq j \leq r$ and K is a given matrix on $\Omega \times R^{2 \times r} $. We characterize a class of matrices K for which we prove error bounds for this discretization. For sufficiently regular solutions ${\bf u}$, achievable at least for some model problems, our bounds improve on existing results in the literature. It is shown that for a notable subclass of K, for which only suboptimal error bounds have been previously derived, the piecewise linear finite element approximation of this problem will converge ...
In this paper we examine the continuous piecewise linear finite element approximation of the following system: given ${\bf f} \equiv (f_j )$ and ${\bf g} \equiv (g_j )$, find ${\bf u} \equiv (u_j )$($(j = 1 \to r$ with $r = 1$ or 2) such that $ - \nabla \cdot (K(z,\nabla {\bf u}(z))\nabla {\bf u}(z)) = {\bf f}(z),\quad z \in \Omega \subset R^2 ,\quad {\bf u} |_{\partial \Omega } = {\bf g} |_{\partial \Omega } ,$ where $(\nabla {\bf u}) \equiv {{\partial u_j } / {\partial z_i }}\, 1 \leq i \leq 2,\, 1 \leq j \leq r$ and K is a given matrix on $\Omega \times R^{2 \times r} $. We characterize a class of matrices K for which we prove error bounds for this discretization. For sufficiently regular solutions ${\bf u}$, achievable at least for some model problems, our bounds improve on existing results in the literature. It is shown that for a notable subclass of K, for which only suboptimal error bounds have been previously derived, the piecewise linear finite element approximation of this problem will converge ...