# Resolution of the 1D Poisson's equation using local Shape Functions

From KratosWiki

- The problem to solve is:

- with the following
**analytical solution**:

- The weak form of the integral equation is:

- and the approach solution is written by using the shape functions as:

## Resolution using 1 element with two nodes

- One single element means to cover the entire domain with the local functions.

- By using the Galerkin method (, it can be obtained:

- for i=1:

- because and

- for i=2:

- because and

- in matricial form:

- with:

- To be more practical, this system of equations can be particularised for some specific polynomial shape functions:

- Therefore:

- is the same value which it has been obtained with the analytical solution. Nevertheless, the global solution is clearly different:

- To obtain the
**reaction**to the fixed value, the first equation of the matricial system of equations can be used:

## Resolution using two elements with two nodes

- Two elements with two nodes each means the use of three global nodes:

- Locally, it is equivalent to:

- for i=1:

- for i=2:

- for i=3:

- in matricial form:

- with:

- In this case:

- If

- and again

- Come back to the Resolution of the Poisson's equation using local Shape Functions