Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L1
Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L1
复制标题
L1 右侧发散型二阶线性椭圆方程的有限元逼近
DOI：
10.1007/s00211-006-0033-2
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发表时间：
2006
影响因子：
2.1
通讯作者：
F. Murat
中科院分区：
文献类型：
--
作者：
J. Casado;T. C. Rebollo;V. Girault;M. G. Mármol;F. Murat
本文考虑了在维数d≥2的情况下，具有L∞上系数的散度型二阶线性椭圆方程（Ω）的标准$$\mathbb{P}_{1}$$有限元逼近，它推广了拉普拉斯方程。我们假设三角剖分族是规则的，并且它满足一个近似于经典假设的假设，它蕴意着离散极大值原理。当右边属于L1（Ω）时，我们证明了离散问题的唯一解收敛于$$W^{1,q}_0(\Omega)$$（对于每一个带有$${1 \leq q < \frac{d}{d-1}}$$的q）到问题的唯一重整化解。当右边是有界Radon测度时，我们得到一个较弱的结果。在维数为d = 2或d = 3且系数是光滑的情况下，当右侧属于Lr（Ω）时，我们在$$W^{1,q}_0(\Omega)$$中给出误差估计。
In this paper we consider, in dimension d≥ 2, the standard $$\mathbb{P}_{1}$$ finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to L1(Ω), we prove that the unique solution of the discrete problem converges in $$W^{1,q}_0(\Omega)$$ (for every q with $${1 \leq q < \frac{d}{d-1}}$$) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon measure. In the case where the dimension is d = 2 or d = 3 and where the coefficients are smooth, we give an error estimate in $$W^{1,q}_0(\Omega)$$ when the right-hand side belongs to Lr(Ω) for some r > 1.