Characterizing Mesh Independent Quadratic Convergence of Newton's Method for a Class of Elliptic Problems
Characterizing Mesh Independent Quadratic Convergence of Newton's Method for a Class of Elliptic Problems
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DOI：
10.1137/100817589
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发表时间：
2012-05
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通讯作者：
J. Karátson
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文献类型：
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作者：
J. Karátson
经典的网格独立原理（MIP）描述了牛顿方法的一个理想性质，其主要特征是离散迭代对任何网格尺寸都表现出相同的二次收敛行为，即随着网格的细化而均匀。研究了一类用有限元离散方法求解的二阶非线性椭圆型边值问题的后一性质。为此，引入了一种更具体的原理，即二次收敛的网格无关原理（MIPQC）。证明了当且仅当椭圆方程为半线性时，MIPQC成立。
The classical mesh independence principle (MIP) describes a desirable property for Newton's method, its main feature being that the discrete iterations exhibit the same quadratic convergence behavior for any mesh size, i.e., uniformly as the mesh is refined. We study the latter property for a general class of second order nonlinear elliptic boundary value problems solved by finite element discretization. For this, a more specific principle, the mesh independence principle for quadratic convergence (MIPQC), is introduced. It is proved that the MIPQC holds if and only if the elliptic equation is semilinear.