Treffer: String gradient weighted moving finite elements

Moving finite element methods are well established for solution of systems of partial differential equations which contain regions where the solution is rapidly varying but moving. The string or second gradient weighted moving finite element method (SGWMFE) uses a piecewise linear discretization of a single evolving manifold to approximate the solution of the PDEs. In the case of one space dimension, x, and two dependent variables, u(x, t) and v(x, t), the solution is calculated from the normal motion of a single manifold [x(τ,t),u(τ,t),v(τ,t)], where τ is a parameter along the maniflold, or a 'string' embedded in [x,u,v] space. This method can be extended to multiple dimensions and an arbitrary number of dependent variables in which case the 'string' parameterization analogy is replaced by a multi-variable parameterization. We outline the application of SGWMFE for solution of the shallow water equations in one and two space dimensions.