On One Approach to Solving a Multiscale Problem With a Kink
This contribution aims to demonstrate a method of application of COMSOL Multiphysics®
to solve a multi-scale problem whose essential feature is a kink with an unknown configuration. The search for its shape and details of its structure is the main task of the study.
This task I studied within the example of a stationary partial differential equation with strong nonlinearity. Its solution is close to zero far from the origin of coordinates and close to a constant value within some not far from the origin. A kink of a complex shape separates these two domains. The lateral size of the area with the nonzero solution is about a few hundred times larger tan the kink width. The latter determines the multi-scale nature of this problem.
To solve the problem, I use the (i) relaxation method. It consists in passing from the static equation - to the dynamic one. The relaxation method makes use of the fact that the solution of the dynamic equation converges to a fixed point. The latter represents the solution of the static equation. Thus for a large enough solution time, t, one can expect to be close to the solution of the static equation.
The relaxation method itself is, however, not enough. To resolve the kink region one should supply it with (ii) the Adaptive Mesh Refinement feature switched on. The Adaptive Mesh Refinement can be tuned to build during the Study several meshes such that in the kink region each next mesh is denser than the previous one while leaving the mesh coarse away from the kink. I did not succeed, however, to get more than ten meshes per study, which is not enough.
The work around is (iii) in generating multiple Studies, each next of them taking the result of the previous one as the initial condition for the new Study. Each of them possesses the Adaptive Mesh Refinement feature.
This strategy brings one to a successful solution of the problem.
