Integration by parts is equivalent to using Green’s function; and so it turns out as follows:

laplacian u = - u + f
Greens function: integ((laplacian u)v) + integ(dot(grad u, grad v)) = surfinteg((partial_n u)v)

So that the weak form is obtained by substituting laplacian u = -u + f from the Helmholtz equation in the Green’s function. The same is arrived at by integration by parts.

Sincerely,

David

> On 10 Dec 2015, at 15:23, Lawrence Mitchell <lawrence.mitchell@imperial.ac.uk> wrote:
>
>
>
> On 10/12/15 14:12, Angwenyi David wrote:
>>>>> nx = 10
>>>>> ny = 10
>>>>> nz = 20
>>>>> mesh = BoxMesh(nx, ny, nz, 1.0, 1.0, 2.0)
>>>>> V = FunctionSpace(mesh, "CG", 1)
>>>>> u = TrialFunction(V)
>>>>> v = TestFunction(V)
>>>>> f = Function(V)
>>>>> f.interpolate(Expression("(2*x[0])+(3*x[1])+(0.5*x[2])"))
>> Coefficient(FunctionSpace(Mesh(VectorElement('Lagrange', tetrahedron,
>> 1, dim=3), 3), FiniteElement('Lagrange', tetrahedron, 1)), 4)
>>>>> a = (dot(grad(v), grad(u)) - v * u) * dx
>
> This is the indefinite form of the Helmholtz equation.  Did you
> actually want the sign-positive Helmholtz equation?
>
> The later has strong form:
>
> -laplacian u + u = f
>
> When integrating by parts the weak form becomes:
>
> <grad u, grad v> + <u, v> = <f, v>
>
> As to incorporation of boundary conditions, see
>
> http://firedrakeproject.org/variational-problems.html#incorporating-boundary-conditions
>
> Cheers,
>
> Lawrence
>
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