The multiscale hybrid mixed method for parabolic problems

Abstract

This thesis aims to generalize the Multiscale Hybrid Mixed method (MHM) for parabolic partial differential equations. This numerical method is based on a primal variational formulation of the problem, where the continuity of the solution on the boundary of the space-time mesh is enforced thru the use of Lagrange multipliers either for space and time. Such approach leads to the formulation of a coupled system of global-local equations, where the solution is the same as the solution of the original problem. The solutions of the local equations turn into a basis used to solve the global problem, and due to the independence of such solutions, they can be numerically approximated in parallel, while capturing the in formation from the fine scales. Since the solutions are obtained thru a time marching scheme, the flexibility of the method reflects on the possibility to use different space-time partitions to approximate numerically the solution on each time interval. Besides, the error estimates for the first level discretization obtained in this work show that the spatial and temporal convergence rates are related to the discretization parameters of the space-time mesh, as well as the degree of the polynomials used to approximate the Lagrange multipliers over the boundary of the mesh.