In this topic, we have introduced the finite element method via the Poisson equations in 1D. Note that the FEM has some advantages compared to the FDM, for example, the mesh is not necessarily uniform. Today, we discuss the question how to apply this FE method to some evolution equations. What we are going to do is to study the heat equations in 2D.
In the literature, the simplest model of the heat equations in 2D over a domain is given as follows
where is a given function. For simplicity, we write . Unlike the heat equations in 1D, we cannot illustrate the solution in just one picture some several . The point is has 3 variables, and thus to achieve a full picture of , we need to draw in 4D, that’s impossible.
Due to the presence of , normally we need to impose an initial condition, i.e., the solution at . The way to solve this equation numerically is to use, for example, the Backward Euler Method. We denote by the difference in time, if we have already known the solution at , then we can solve at . The Backward Euler Method applied to this model equation gives
Usually, for the sake of clarity by we mean we are working on the -th step, therefore, at -step, the time is given by , the solution is then denoted by which is actually . The time-discretization equation is then given by
We assume is a triangulation of , that means, we devide the domain into small special triangles. Vertices of triangles, usually called nodes, are those points we want to approximate . If we denote by the finite-dimensional subspace of with a basis such that for each , the restriction of into is piecewise linear in the sense that is linear on each triangle of .
The above picture shows us a case when is a ball. The idea of FEM is to approximate solution by some such that for every vertices . Let talk about the basis . Functions at the -node are chosen so that and for every .
We are now interested in finding weak solutions to the heat equations. In this situation, one tries to find a function which satisfies the following equation
for every .
we then have
If we apply the above equation for being , we then have
In term of linear algebra, the above equations can be transfered to a linear system where
The rest of our task is to solve this system of linear equations.
Source: http://en.wikipedia.org/wiki/Finite_element_method and a book due to Claes Johnson.