ABSTRACT
The moving finite element method described in Chapter 2 has been extensively tested for the numerical solution of convection-diffusion-reaction model in 1D spatial domains in Chapter 3. This chapter deals with the application of the moving finite element method to the two-dimensional spatial domains reported in Section 2.3.1. The implementation of the 2D MFEM algorithm uses a single spatial mesh to discretize the regular (rectangular) spatial domain even in the case of a system of PDEs. Also, a simplified set of penalty constants is used, and it is assumed that they are equal for all finite elements. The experience from the application of the 1D MFEM to different problems supports this option since the penalty constants are introduced only to prevent singularities of the matrix defining the system of ODEs obtained from the spatial discretization. Let us consider a finite element, Ωi, of area A(Ωj). The simplified version of Equation 2.40 is defined by the equations,
Si = c1
A(Ωi)− c3
( 1 +
c3 A(Ωi)− c3
)2 (4.1)
i =
( c2
A(Ωi)− c3 + c2 )(
1 + c3
A(Ωi)− c3
)2 (4.2)
where ci, i = 1, 2, 3 are small positive constants supplied by the user. The constant c3 represents the minimum allowed for an element area. The users of the 2D MFEM algorithm must also choose the degree for Gaussian quadrature formulae to be used in numerically solving integrals over 1D or 2D domains. The finite elements are triangles defined from a logically rectangular mesh by adding diagonals. Thus the same set of spatial nodes can produce two different meshes, depending on the orientation of the diagonals. By default, nodes are placed equally spaced, and the user may define the number of nodes in both axes. Figure 4.1 illustrates both cases for an equally spaced mesh with 72 triangles, i.e., 49 nodes over Ω = [0, 1]× [0, 1].
