Collocation and finite element methods

Both the collocation and the finite element methods are function approximation methods. Similar to the finite difference method, the strategy here is to reduce the differential equation to a set of algebraic equations that can be solved. Instead of discretizing the differential equation by replacing the various derivatives with difference-quotient approximations, the solution is given a functional form. 

In other words, the function approximation methods find a solution by assuming a particular type of function, a trial (basis) function, over an element or over the whole domain, which can be polynomial, trigonometric functions, splines, etc. These functions contain unknown parameters that are determined by substituting the trial function into the differential equation and its boundary conditions. In the collocation method, the trial function is forced to satisfy the boundary conditions and to satisfy the differential equation exactly at some discrete points distributed over the range of the independent variable, i.e. the residual is zero at these collocation points. In contrast, in the finite element method, the trial functions are defined over an element, and the elements, are joined together to cover an entire domain. 

The collocation method is a simple and effective method that is illustrated in the following example. Consider solving the following BVP problem by using the collocation method  

In this case, we will select a trial function, (pi x), which can fulfill the boundary conditions 

Three equations are obviously required to determine the unknowns a, a2,az. By substituting Equation (6.65) into the differential equation Equation (6.63) and evaluating at certain mesh points, the problem is reduced to solving a system of equations in ai. The collocation points should be selected at points that have a large influence on the function. We require that the BVP is satisfied in the middle of the domain, at the collocation point X2 = 0.5, which yields 

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A differential equation that has data given at more than one value of the independent variable is a boundary-value problem (BVP). Consequently, the differential equation must be of at least second order. The solution methods for BVPs are different compared to the methods used for initial-value problems (IVPs). An overview of a few of these methods will be presented in Sections 6.2.1. 2.3. The shooting method is the first method presented. It actually allows initial-value methods to be used, in that it transforms a BVP to an IVP, and finds the solution for the IVP. The lack of boundary conditions at the beginning of the interval requires several IVPs to be solved before the solution converges with the BVP solution. Another method presented later on is the finite difference method, which solves the BVP by converting the differential equation and the boundary conditions to a system of linear or non-hnear equations. Finally, the collocation and finite element methods, which solve the BVP by approximating the solution in terms of basis functions, are presented. 

What do the collocation and finite element methods have in coimnon, and how do they differ ... 

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