Project:

Inhomogeneous mesh with FEMs.

Specification of project  Inhomogeneous mesh with FEM's .
 

1. Construct the inhomogeneous mesh.


The inhomogeneous mesh is constructed from a grid density function fg(x), that is the sum of a number N Lorentzian functions

and a constant function. The integral of fg(x)

is the relative number of gridpoints xi, such that x>xi.

This has been inplemented into  mesh.java ( see the functions  gridfunction and  Mesh ).
 
 

2. Calculate the matrix elements using a 2-points Gaussian quadrature.

To calculate the matrix elements, the basis function, it's derivative as well as the velocity and diffusion coefficients as functions of x, see FEMSolution.java, row  27-72 .

The linear system is

The matrix coefficients in Latex notation 

A_{ij} = \int_{0}^{x_{max}} dx  [ e_i (x) e_j(x)  +  ( u(x) e_i (x) e'_j(x) + D(x) e'_i (x) e'_j(x) )  \theta \Delta t]
B_{ij} = \int_{0}^{x_{max}} dx  [ e_i (x) e_j(x)  -  ( u(x) e_i (x) e'_j(x) + D(x) e'_i (x) e'_j(x) )  \theta \Delta t]
where $ \theta  \in [ 0.5 , 1 ] $ is a numerical parameter (describing the expliciteness), $ u , D $ is the velocity and the diffusion coefficient and $ e' =de/dx $. The integrals are evaluated using a 2-point Gaussian quadrature (see lecture notes). This has been inplemented into FEMSolution.java 149-220 .

Before investigating a problem with an inhomogeneous mesh it was confirmed the new implementation of the numerics gave the same result as the original implementation.

3. Study the numerical convergence of an advection-diffusion problem in a strongly inhomogeneous medium.

I have studied a mesh where the grid density is the sum of two Lorentzian functions, with  $ x1 = 0.2, w1 = 0.05 $ and $ x2 = 0.6, w2 = 0.1 $. On this mesh the I've studied a soliton puls propagated by the advection equation. The puls seems to propagate well where the mesh is dense. However, when the grid density is low the numerical dispersion changes the shape of the puls.When the puls enters a region with low grid density, short wavelehgths are reflected and long wavelengths are being damped. This could be caused by the problem of representing the pulse when the grid density is low, which reflects the waves with wave lengths shorter then the local grid spacing. However, the total moment of the function is conserved  (deviation of 5e-6).

Please run (use the "FEM / Tunable Integration")
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