C++ for Scientists - Technische Universität Dresden

How to Handle Physics on the
Computer
Chapter 9
9.1 Finite Elements
Discretization schemes lead in general to a linear system of equations:
These matrices are typically:
• sparse (there are only few non-zero elements per row)
• large dimension N (10 4 − 10 9 unknowns)
—xx
A x = f (9.1)
The non-zero elements of the matrix Ai,j represent a finite element with both degrees of freedom
i and j connected.
To demonstrate the transfer of a continuous formulated equation such as the Laplace or Poisson
equation to the finite regime of a computer, a simple Dirichlet problem is used. If an implicit
(uniform) 1D-grid with n elements is used, the contribution of each element to the system
matrix A is constant, so called stencil sub-matrix.
⎛
⎜
A = ⎜
⎝
2 −1
−1 2 −1
2D implicit grid of dimension N = (n − 1) 2 is:
−1 2 −1
−1 2
233
⎞
⎟
⎠
(n−1)x(n−1)