Finite Element Adaptation for Pdes: Duality Methods for Time-space Adaptivity

Nowadays, numerical methods are crucial in order to understand the physics behind great amount of industrial processes. Engineers, mathematicians, physicists and scientists in general, work on a wide variety of problems modelled by partial differential equations (PDEs). In most of these problems the physical phenomena are extremely complex, with several scales of time and space, making difficult to calculate the numerical solution of the equations. In this work we propose an efficient numerical treatment of time dependent PDEs equations, with special interest on fluid dynamics and combustion problems, using adaptive techniques. The goal of our adaptation is to design both economical meshes and time steps sizes to compute an accurate solution with a reduced computational cost. The adaptation will be based on information extracted from an a posteriori error estimator calculated by duality techniques in terms of an output target functional, J(u), of the solution u of our problem.