AI 中文总结
研究具有混合边界条件的拟线性椭圆热传导问题，用牛顿法求解伽辽金离散化产生的非线性代数系统，结合混合边界椭圆正则性与对偶论证，证明离散解满足最优收敛速率，数值实验证实收敛速率及准则可行性。
AI 中文摘要
本文研究了具有非齐次混合边界条件的拟线性椭圆热传导问题的有限元逼近。电导率张量是矩阵值的、各向异性的、可能非对称的，并且依赖于位置和温度，这使得问题具有非线性、非单调性和非势性。通过牛顿法求解由伽辽金离散化产生的非线性代数系统，并由可计算的牛顿 - 康托罗维奇准则和与网格相关的停止规则提供后验保证，确保代数误差相对于离散化误差渐近可忽略。由于离散解不一定唯一，我们证明了每个离散解都满足最优收敛速率。该分析将混合边界椭圆正则性与适用于拟线性设置的奥宾 - 尼茨切对偶论证相结合。二维和三维的数值实验证实了预测的收敛速率，并证明了所提出准则的可行性。
英文摘要
The article studies finite element approximations of a quasilinear elliptic heat-conduction problem with inhomogeneous mixed boundary conditions. The conductivity tensor is matrix-valued, anisotropic, possibly nonsymmetric, and dependent on both position and temperature, rendering the problem nonlinear, nonmonotone, and nonpotential. The nonlinear algebraic system arising from the Galerkin discretization is solved using Newton's method, with a posteriori guarantees provided by a computable Newton-Kantorovich criterion and a mesh-dependent stopping rule that ensures that the algebraic error is asymptotically negligible relative to the discretization error. Since the discrete solution need not be unique, we prove that every discrete solution satisfies the optimal convergence rates. The analysis combines mixed-boundary elliptic regularity with an Aubin-Nitsche duality argument adapted to the quasilinear setting. Numerical experiments in two and three dimensions confirm the predicted convergence rates and demonstrate the feasibility of the proposed criterion.
