# Polynomial Fitting

• To introduce the idea of finding coefficients to functions, let's consider simple polynomial fitting.

• In polynomial fitting (or interpolation) you have a set of points and you are looking for the coefficients to a function that has the form:

• Where a, b and c are scalar coefficients and 1, x, x^2 are "basis functions".

• Find a, b, c, etc. such that f(x) passes through the points you are given.

• More generally you are looking for: where the c_i are coefficients to be determined.

• f(x) is unique and interpolary if d+1 is the same as the number of points you need to fit.

• Need to solve a linear system to find the coefficients.

# Example

• Define a set of points:

• Substitute data into the model:

• Leads to the following linear system for , , and :

• Solving for the coefficients results in:

• These define the solution function:

• Important! The solution is the function, not the coefficients.

• The coefficients themselves don't mean anything, by themselves they are just numbers.

• The solution is not the coefficients, but rather the function they create when they are multiplied by their respective basis functions and summed.

• The function does go through the points we were given, but it is also defined everywhere in between.

• We can evaluate at the point , for example, by computing: or more generically: where the correspond to the coefficients in the solution vector, and the are the respective functions.

• Finally, note that the matrix consists of evaluating the functions at the points.

# Finite Elements Simplified

• A method for numerically approximating the solution to Partial Differential Equations (PDEs).

• Works by finding a solution function that is made up of "shape functions" multiplied by coefficients and added together.

• Just like in polynomial fitting, except the functions aren't typically as simple as (although they can be).

• The Galerkin Finite Element method is different from finite difference and finite volume methods because it finds a piecewise continuous function which is an approximate solution to the governing PDE.

• Just as in polynomial fitting you can evaluate a finite element solution anywhere in the domain.

• You do it the same way: by adding up "shape functions" evaluated at the point and multiplied by their coefficient.

• FEM is widely applicable for a large range of PDEs and domains.

• It is supported by a rich mathematical theory with proofs about accuracy, stability, convergence and solution uniqueness.

# Weak Form

• Using FE to find the solution to a PDE starts with forming a "weighted residual" or "variational statement" or "weak form". - We typically refer to this process as generating a Weak Form.

• The idea behind generating a weak form is to give us some flexibility, both mathematically and numerically.

• A weak form is what you need to input to solve a new problem.

• Generating a weak form generally involves these steps:

1. Write down strong form of PDE.

2. Rearrange terms so that zero is on the right of the equals sign.

3. Multiply the whole equation by a "test" function .

4. Integrate the whole equation over the domain .

5. Integrate by parts (use the divergence theorem) to get the desired derivative order on your functions and simultaneously generate boundary integrals.

# Refresher: The divergence theorem

• Transforms a volume integral into a surface integral:

• Slight variation: multiply by a smooth function, :

• In finite element calculations, for example with , the divergence theorem implies:

• We often use the following inner product notation to represent integrals since it is more compact:

• http://en.wikipedia.org/wiki/Divergence_theorem

# Example: Convection Diffusion

• Write the strong form of the equation:

• Rearrange to get zero on the right-hand side:

• Multiply by the test function :

• Integrate over the domain :

• Apply the divergence theorem to the diffusion term:

• Write in inner product notation. Each term of the equation will inherit from an existing MOOSE type as shown below.