We prove linear convergence for a new family of modified Dirichlet--Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin--Robin method. Such nonoverlapping domain decomposition methods are commonly employed for the parallelization of partial differential equation solvers. Convergence has been extensively studied for elliptic equations, but in the case of parabolic equations there are hardly any convergence results that are not relying on strong regularity assumptions. Hence, we construct a new framework for analyzing domain decomposition methods applied to quasilinear parabolic problems, based on fractional time derivatives and time-dependent Steklov--Poincar\'e operators. The convergence analysis is conducted without assuming restrictive regularity assumptions on the solutions or the numerical iterates. We also prove that these continuous convergence results extend to the discrete case obtained when combining domain decompositions with space-time finite elements.

翻译：我们证明了应用于拟线性抛物型方程的一类新修正Dirichlet-Neumann方法的线性收敛性，以及Robin-Robin方法的收敛性。这类非重叠区域分解方法常用于偏微分方程求解器的并行化。虽然椭圆型方程的收敛性已得到广泛研究，但在抛物型方程情形下，几乎不存在不依赖强正则性假设的收敛性结果。为此，我们基于分数阶时间导数与时变Steklov-Poincaré算子，构建了分析拟线性抛物型问题区域分解方法的新框架。该收敛性分析无需对解或数值迭代施加限制性正则性假设。我们还证明了这些连续收敛结果可推广至区域分解与时空有限元结合所得到的离散情形。