Fractal Interfaces and Heat Transmission Problems

The main portion of my thesis focuses on a 2-dimensional second order heat transmission problem in domains with pre-fractal interfaces. My focus is on the numerical approximation of the solutions. Precisely, I€™m concerned to develop a suitable mesh refinement algorithm that could be adapted to our situation, by taking into account the regularity of the solutions and the geometry of irregular pre-fractal interfaces. I obtain an error estimate between the weak solution and the discrete solution, which indicates an optimal rate of convergence as in the classical case when the solution has H^2-regularity. In addition, numerical simulations are also included, which demonstrates the features of our heat transmission model. Another portion of my thesis focuses on the asymptotic analysis of singular boundary value problems with highly conductive layers of pre-fractal type. My models illustrate the problems of a lower- dimensional highly conductive material intruding into a higher- dimensional material with lower conductivity. I consider a 2D model of Sierpinski pre-fractal layers and 3D models of hierarchical layers. The main results consist in the so-called Mosco-convergence of certain energy functionals, which implies the strong convergence of the solutions and of the spectral resolutions as a byproduct in real applications