1D Poisson

The 1D Poisson equation with Dirichlet boundary and smooth functions is about as simple as it gets. This example demonstrates the basics of setting up a problem in Finch. A uniform discretization of the unit domain is used with p=4 polynomial basis function space.

Begin by loading the Finch module. Then initialize. The name here is only used when generating code files.

using Finch
initFinch("poisson1d");

Then set up the configuration. This example simply sets dimensionality of the domain and polynomial order of the basis function space.

domain(1)                  	# dimension
functionSpace(order=4) 		# polynomial order

Use the built-in simple mesh generator to make the mesh and set up all node mappings.

mesh(LINEMESH, elsperdim=20)# uniform 1D mesh with 20 elements

Define the variable, test function, and forcing function symbols.

u = variable("u")           # make a scalar variable u
testSymbol("v")             # sets the symbol for a test function
coefficient("f", "-100*pi*pi*sin(10*pi*x)*sin(pi*x) - pi*pi*sin(10*pi*x)*sin(pi*x) + 20*pi*pi*cos(10*pi*x)*cos(pi*x)")

Convert the PDE

$\Delta u=f(x)$

$u(0)=u(1)=0$

into the weak form

$-(\nabla u,\nabla v)=(f,v)$

The boundary condition is specified.

boundary(u, 1, DIRICHLET, 0) # boundary condition for BID 1 is Dirichlet with value 0

Then write the weak form expression in the residual form. Finally, solve for u.

weakForm(u, "-grad(u)*grad(v) - f*v")
solve(u);

End things with finalizeFinch() to finish up any generated files and the log.