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: (1)

  • Where , and are scalar coefficients and , , $ are "basis functions".

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

  • More generally you are looking for: (2) where the are coefficients to be determined.

  • is unique and interpolary if 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: (3)

  • Substitute data into the model: (4)

  • Leads to the following linear system for , , and : (5)

  • Solving for the coefficients results in: (6)

  • These define the solution function: (7)

  • 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: (8)

  • Slight variation: multiply by a smooth function, : (9)

  • In finite element calculations, for example with , the divergence theorem implies: (10)

  • We often use the following inner product notation to represent integrals since it is more compact: (11)

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

Example: Convection Diffusion

  • Write the strong form of the equation: (12)

  • Rearrange to get zero on the right-hand side: (13)

  • Multiply by the test function : (14)

  • Integrate over the domain : (15)

  • Apply the divergence theorem to the diffusion term: (16)

  • Write in inner product notation. Each term of the equation will inherit from an existing MOOSE type as shown below. (17)