The implicit Euler scheme for nonlinear partial differential equations of gradient flows is linearized by Newton’s method, discretized in space by the finite element method. With two Newton iterations at each time level, almost optimal order convergence of the numerical solutions is established in both the |$L^q(\varOmega )$| and |$W^{1,q}(\varOmega )$| norms. The proof is based on techniques utilizing the resolvent estimate of elliptic operators on |$L^q(\varOmega )$| and the maximal |$L^p$|-regularity of fully discrete finite element solutions on |$W^{-1,q}(\varOmega )$|⁠.

中文翻译：

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牛顿法线性化梯度流的有限元方法

梯度流非线性偏微分方程的隐式Euler方案通过牛顿法线性化，在空间中通过有限元法离散化。通过在每个时间级别进行两次牛顿迭代，可以在| $ L ^ q（\ varOmega）$ |中建立数值解的几乎最优阶收敛。和| $ W ^ {1，q}（\ varOmega）$ | 规范。该证明基于利用| $ L ^ q（\ varOmega）$ |上的椭圆算子的分解估计的技术。和最大值| $ L ^ p $ | | $ W ^ {-1，q}（\ varOmega）$ |⁠上的完全离散有限元解的正则性。