Equation of Interest

1
Equation of Interest
After application of the method of weighted
residuals, the equation of interest is
2
2-D Conservation of Mass
Let us consider a single element,
3
2-D Conservation of Mass (contd 2)
In vector notation,
4
Method of Weighted Residuals
5
Method of Weighted Residuals (contd 2)
From the Divergence Theorem,
The inter-element integrals cancels out
everywhere except on the outer boundary of the
grid.
6
Method of Weighted Residuals (contd 3)
Approximation of variables assuming bilinear
quadrilateral elements (4 node elements)
7
Method of Weighted Residuals (contd 4)
Rearranging terms,
8
Conservation of Momentum
In vector form,
9
Local Matrix Build

• The local matrix associated with the
  conservations of mass and momentum contains three
  sets of unknowns Dr,Du, and Dv.
• The local matrix is 12x12 (3 unknowns at each of
  4 nodes for bilinear quadrilateral elements).

10
Global Matrix Build

• The resulting global matrix is sparse, but, in
  general, is not banded like in the quasi-1D case.
• The solution of the matrix equation can be done
  either by direct or iterative solver
• Direct Computation time proportional to N1.5
  where N is the number of nodes in the grid.
  Storage proportional to NlogN.
• Iterative Computation time is proportional to aN
  where a is the number of iterations necessary for
  convergence. Storage proportional to N.

11
Direct vs. Iterative Solvers for FEM

• Direct solver may be more efficient for linear
  problems (1 LU decomposition, many right hand
  sides).
• Iterative solvers usually more efficient if
  solver converges (usually very few iterations for
  convergence in time domain methods).
• In 3-D problems with large number of unknowns,
  direct solvers may be infeasible (computation
  time grows to N2 and storage also grows).
• For parallel computations, the iterative solvers
  are more efficient.

12
Comparison of FEM to FD ADI

• FD ADI will almost always be more efficient for
  solving 2-D and 3-D problems.
• The inherent FD ADI algorithm limits the type of
  grids that can be used. Use of the structured
  grids in FD (and especially when coupled to ADI)
  can result in significant errors in the solution.
• The flexibility of FEM to handle unstructured
  grids is very important especially for 3-D
  problems.