String Gradient Weighted Moving Finite Elements for systems of Partial Differential Equations I

Abstract

Moving finite element methods resolve regions containing steep gradients using a manageable number of moving nodes. One such implementation is Gradient Weighted Moving Finite Elements (GWMFE). When applied to a single PDE with one space variable x, the solution u(x,t), is viewed as an evolving parameterized manifold. Miller (1997) derived the ''normal motion'' of the manifold in [x,u] space and discretised in space by making the manifold piecewise linear. For systems of PDEs, he used a separate manifold for each dependent variable but with shared nodes. However, Miller also proposed a ''second GWMFE formulation for systems of PDEs''. In the case of two dependent variables u(x,t) and v(x,t), instead of determining the separate normal motion of two manifolds, using shared nodes, he suggested examining the normal motion of a single manifold, a ''string'' embedded in [x,u,v] space . This method, called String Gradient Weighted Moving Finite Elements (SGWMFE), has not previously been implemented and tested. In this paper we revisit the SGWMFE method, deriving a general form of the equations for normal motion using a projection matrix and implementing the method for the one dimensional shallow water equations and for Sod's shock tube problem.