Next: The input file Up: The PDE Description file Previous: The PDE Description file Contents

# The weak formulation

The partial differential equation describing a heat flow problem with thermic conductivity , a volumetric source density , an inflow density at the part of the boundary, and a heat exchange to the environment of temperature , and conductivity on is

The weak form of the equation is to find the temperature distribution in the proper Sobolev space such that

holds for all test functions . The finite element method replaces the Sobolev spaces by a finite dimensional sub-space.

The PDE input file defines the variational problem. You specify the finite element space, and then, the bilinear-form (left hand side), and the linear form (right hand side) are build up from elementary blocks called integrators. The table below lists the names of the integrators needed in the equation above:

> laplace lam |
> robin alpha |
> source f |
> neumann g |

Note that the last term is handled by the neumann - integrator with coefficient . You might miss Dirichlet boundary conditions. Indeed, NGSolve always approximates them by Robin b.c. with large conductivity .

Next: The input file Up: The PDE Description file Previous: The PDE Description file Contents