Being able to solve numerically partial differential equations is fundamental for engineers to evaluate, optimize and improve industrial equipments. The framework of mimetic finite element methods allows engineers to find solutions characterized by strong conservation properties: this may result in a pointwise divergence free-flow field. However, sometimes, the computation of the solution of partial differential equations is time consuming, as a results, to reduce the computational time, engineers and mathematicians have developed hybrid methods.
The objective of this thesis is the development of a hybrid mixed finite element formulation of the vector Laplace equation without spurious modes. Discontinuous elements permit an higher degree of parallelism, and, at the end, a lower computational time. Lagrange multipliers are used to impose continuity between discontinuous elements. These turn out to be not only mathematical features but they are connected to the physical variables of the problem. Furthermore, it has been found that the usage of a new Lagrange multiplier, on the intersection of 4 or more elements, removes the spurious modes. Therefore, the associated system of equation is non-singular. The usage of the hybrid finite element methods reduces the computational time while maintaining the pointwise divergence constraint and the optimal convergence rate of all variables.
At the end, the mixed hybrid formulation is modified to solve the Stokes equations. Lagrange multipliers are used as boundary conditions. Solution of the lid-driven Stokes flow is shown.