- مبلغ: ۸۶,۰۰۰ تومان
- مبلغ: ۹۱,۰۰۰ تومان
Phase field equations provide a flexible mathematical tool to describe the evolution of interfaces or surfaces in various processes such as crystal growth, multiphase flows, or crack propagation. In contrast to sharp interface models their numerical implementation can be realized with standard methods and they are capable of describing topological changes effectively. The simplest example is the Allen–Cahn equation ∂tu − 1u = −ε −2 f (u) in which ε > 0 is a small parameter that describes the thickness of the diffuse interface that separates regions in which u ≈ ±1 and f is the derivative of a double well potential, e.g., f (u) = 2(u 3 − u). Fig. 1 displays snapshots of a simple but generic evolution leading to a generic topological change, i.e., a circular interface shrinks and disappears in finite time. Although the mathematical modeling of such events is unclear the agreement with experiments is quite remarkable. The topological change corresponds to a singularity in the evolution and the approximation properties of numerical methods may be critically affected. It is the aim of this article to provide a theoretical justification for the reliability of standard numerical methods at topological changes in a simple model situation.
We have discussed in this paper the robustness of error estimates for the approximation of phase field models with standard numerical techniques. Those error estimates avoid an explicit exponential dependence on the inverse of the small phase field parameter but include the principal eigenvalue of the linearized differential operator. The precise properties of this crucial quantity are only rigorously understood for the smooth evolution of interfaces. Numerical experiments reveal a scaling behavior at singularities that implies the robustness of the error estimates through topological changes. For an important class of generic topological changes we have shown that this behavior can be rigorously analyzed and thereby explained the surprisingly good approximation properties of standard computational methods at singularities.