Finite element method linear interpolation

The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-80) to provide the Galerkin finite element equations. For example, with the grid shown in Fig. 3-48, a linear interpolation would be used between points x, and, vI+1. 

The functions in the finite element method are constant, linear, or quadratic functions of position in a small region (called a finite element). The idea is explained for interpolation of a known function, Eq. (F.31) ... 

For the spatial discretization, the Finite Element Method is employed. To be more specific, the same spatial interpolation basis has been selected for the whole set of primary variables as well as for their variations, albeit this is not required. In case of linearly elastic constitutive equations, and since variations of the primary variables are independent, the following discrete system of equations is obtained ... 

Let be a well-defined finite element, i.e. its shape, size and the number and locations of its nodes are known. We seek to define the variations of a real valued continuous function, such as/, over this element in terms of appropriate geometrical functions. If it can be assumed that the values of /on the nodes of Oj, are known, then in any other point within this element we can find an approximate value for/using an interpolation method. For example, consider a one-dimensional two-node (linear) element of length I with its nodes located at points A(xa = 0) and B(a b = /) as is shown in Figure 2.2. 

The Galerkin method is applied with linear finite elements. The weak formulation of Eqs. (la) and (lb) is obtained by taking simultaneously the products of the equations with appropriate test functions and integration by parts of the spatial derivatives. We use a Lagrangian interpolation of the approximate solutions C for the aqueous solute concentration C, and S for the sorbed phase concentration Si for every species ... 

Boundary elements are based on the use of interpolating functions whose volume integration reduces to the integration at the surfaces. In this way, the number of elements and of nodes/ degrees of freedom is considerably reduced. One of the drawbacks of the boundary element method is that it gives rise to fully populated matrices. Real size of the matrix tends to rise with the square of the size of the problem, while in finite element, the size tends to rise linearly with the size of the problem. 

To accomplish this one needs to parameterize flic solution in terms of a finite set of solution values and then seek to minimize the functional with respect to the selected set of solution values. The set of solution values are taken as some set over each triangular element. The simplest sets of solution values to select are the three node point values as shown in Figure 13.2(a). However, a more complex set could be the six or ten values shown in Figure 13.2(b) and 13.2(c). After selecting the parameterizing set of values an interpolation method is then needed over the spatial domain. The conventional approach is to assume that the solution varies linearly between the triangular nodes and thus to use the shape function as derived in the previous section to approximate the spatial variation of the solution. When written in this form the solution becomes ... 

---