ABSTRACT
The classical Finite Element Method (FEM) is a very pervasive and important approach to find the solution of Partial Differential Equations (PDEs). In the classical FEM, the approximate solution to PDEs is obtained by the discretization of the spatial domain into continuous volume/area elements. The local approximations for each element are obtained by the use of polynomial basis functions, which interpolate the solution and satisfy additional constraints, like exact interpolation at nodal points and inter-element continuity conditions. Generally, the polynomial degree, p, is fixed, and is dictated by the element type and the number of nodal points per element. The success of the classical FEM depends upon the approximation ability of the polynomial basis functions, and further improvement in the approximated solution can be achieved, for a given set of interpolation functions, only by refining the mesh size, h. In most cases, the degree of these basis functions is less than or equal to 2. For example, in the case of the triangular mesh with three nodes per element one can only use degree one polynomials for interpolation to satisfy necessary continuity requirements. Further, in some problems the use of non-polynomial basis functions may be desirable to achieve better accuracy. For example, in the case of the Helmholtz equation, the solution is known to be highly oscillatory in nature, and therefore, it may be desirable to use nonpolynomial basis functions to approximate the exact solution. However, even though the analytical knowledge about local behavior of an exact solution is available, there are no convenient means to incorporate this knowledge in the conventional FEM solution. Also, the reliance of the conventional FEM on a mesh is not well suited to problems involving discontinuities and moving domains. Generally, to deal with moving domains and discontinuities in the conventional FEM methods, the original mesh is regenerated in each step of
the evolution so that mesh lines are in accordance with the moving domain and discontinuities. However, this strategy of re-meshing at each stage can introduce numerous difficulties, such as the need to project the solution between meshes in successive stages of the problem, complexity in the computer program, and of course the computational burden associated with a large number of re-meshing steps.
