Finite element analysis of a convection-diffusion equation
When a solid object interacts with a flowing medium (such as, e.g., water or air), the molecules of the fluid change their velocity very rapidly within a thin film adjacent to the object's surface. This film is called boundary layer. Due to the extremely small width of boundary layers, their numerical approximation is challenging. In particular, due to the hyperbolic nature of the equations of fluid dynamics, numerical errors committed in the boundary layer are transported quickly into the rest of the computational domain, and they may spoil the results of the computation completely. Therefore, accurate and efficient approximation of flows in boundary layers is a topic of paramount importance in aerospace, air force, and naval research. In this work we analyse a model linear convection-diffusion equation which exhibits a boundary layer, and study the optimality of meshes for its finite element approximation. So far, the best meshes available are the Shishkin and Bakhvalov meshes. We construct a new class of meshes which are based on the equidistribution of the piecewise-linear interpolation error in the finite elements. Numerical results show that such meshes have better approximation properties than both the Shishkin and Bakhvalov meshes. ^
Avila, Jose, "Finite element analysis of a convection-diffusion equation" (2006). ETD Collection for University of Texas, El Paso. AAI1435322.