Linearly convergent nonoverlapping domain decomposition methods for quasilinear parabolic equations. (English) Zbl 07930658
Summary: 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-Poincaré 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.
MSC:
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 65J08 | Numerical solutions to abstract evolution equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
nonoverlapping domain decompositions; quasilinear parabolic equations; linear convergence; time-dependent Steklov-Poincaré operators; space-time finite elementsReferences:
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