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# The input file

The file below is a valid input file to NGSolve. The first two lines load the prepared geometry file and the mesh file. It is a square. The boundary splits into the 4 sides to . We specify Robin boundary conditions (with conductivity ) on to , and Neumann boundary condition (with ) on . This is defined in terms of coefficients. The list of numbers correspond to the sub-domains, if the integral is taken over the domain, or, to parts of the boundary, if the integral is taken over the boundary, respectively. The next lines define the mathematical objects finite-element space, grid-function, bilinear-form, linear-form, and a preconditioner. The last line (numproc) calls the solver for boundary value problems (bvp). Here, a preconditioned conjugate gradients iteration is called.

    geometry = ngsolve/pde_tutorial/square.in2d
    mesh = ngsolve/pde_tutorial/square.vol

    define coefficient coef_lam
    1,

    define coefficient coef_alpha
    1e5, 1e5, 1e5, 0,

    define coefficient coef_f
    1,

    define coefficient coef_g
    0, 0, 0, 1,

    define fespace v -order=1
    define gridfunction u -fespace=v -nested

    define bilinearform a -fespace=v
    laplace coef_lam
    robin coef_penalty

    define linearform f -fespace=v
    source coef_source
    neumann coef_g

    define preconditioner c -type=multigrid -bilinearform=a -smoothingsteps=1

    numproc bvp np1 -bilinearform=a -linearform=f -gridfunction=u -preconditioner=c \
        -maxsteps=50