Moving finite elements method applied to the solution of front reaction models: causticizing reaction

Abstract

In this paper a Moving Finite Elements Method based on cubic Hermite polynomials is presented. PDE's solution is obtained by minimizing the square norm of discretized residuals all over the domain in order to the time derivatives of the solutions on the nodes and nodal velocities. Spatial discretization gives origin to a system of ODEs, that can degenerate when singularities occur. These can be of two kinds: parallelism and nodal coalescence. Nodal coalescence is avoided by fixing the minimum internodal distance while parallelism is dealt with the addition of penalty functions to the objective function resulting from the application of Galerkin Method. Boundary conditions are previously time derived originating constraints to the minimization system arising from the basic formulation. The solution of this problem is obtained via the Lagrange Multipliers Method. In this paper this methodology is applied to the solution of a front reaction model - the causticizing reaction. The results obtained are in close agreement with those published in the literature where orthogonal collocation in Finite Elements was used. The adequacy of the Moving Finite Elements Method for the solution of front reaction models is demonstrated in this paper.