Diverse problems of physics and engineering are governed by partial differential equations in space and time. Real-world problems also involve very complicated geometrical objects such as airplanes, automobiles, trains, bridges, buildings, machines, semiconductor devices, the human anatomy, etc. The finite element method was the first general approach developed for solving broad classes of partial differential equations on complicated geometries of a real-world nature. Despite this generality, and the success of the finite element method on a variety of problem classes, shortcomings have been noted in several areas of considerable practical importance, most notably fluid mechanics.
The origin of the problem is with the variational formulation that has been the generally accepted basis of finite element discretizations, that is, Galerkin's method. This method is a member of the class of so-called weighted residual methods and it possesses optimal properties of approximation for specific symmetric, positive-definite operators such as those arising in heat conduction and elasticity. However, when applied to problems in which the operators are dominantly skew-symmetric, such as those arising in many fluid mechanical applications, the optimal properties of Galerkin's method are lost, and often very poor approximation properties are observed.
It has been noted that the deficiency of Galerkin's method in fluid mechanical applications can be traced to its poor numerical stability behavior for skew-symmetric operators. Classical ad hoc solutions based on artificial viscosity and upwinding techniques improve stability but seriously degrade accuracy. For many years the fundamental open problem in computational fluid mechanics was to develop procedures which combined good stability and accuracy properties. A general solution to this problem was achieved by early renditions of what has become to be known as "stabilized methods." The theory of stabilized methods begins with Galerkin's method, but incorporates certain additional terms that preserve the weighted residual format, thus retaining the underlying accuracy, while at the same time enhancing stability. The end result is a robust and general procedure applicable to the widest variety of physical problems including, but not limited to, fluid mechanics.
In my lecture I will present an overview of the fundamental ideas.