Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem. (English) Zbl 1307.65123
Summary: We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the \( L^2(L^2)\) error of the temperature and the \( L^2(H^{-1})\) error of the enthalpy. Moreover, they allow us to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.
MSC:
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 80A22 | Stefan problems, phase changes, etc. |
| 35R25 | Ill-posed problems for PDEs |
| 35K55 | Nonlinear parabolic equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
Keywords:
finite volume method; time-dependent two-phase Stefan problem; a posteriori error estimates; regularization; linearization; algorithm; mesh refinement; Newton method; numerical resultReferences:
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