Using 2d tools tools for 1d problems

To solve the basic Laplace problem with homogeneous boundary conditions on \( (0,1)\) use 2d tools to construct the 1d problem.

Set a 2d mesh Th using a line of triangles sitting on \((0,1)\). This gives our first uniform partition on \((0,1)\) labelled by 1

All we need will be generated and stored in 2d structures and extracted from them.

• Given Th, we can extract the dofs on label 1 constructing one pointer to them zoi, see the example 2 in Quick and dirty tips
• two variable functions u(x,y) on label 1 are the one variable functions u(x,0) that we need
• Using varf mat(u,v)= int1d(Th,1)(dx(u)*dx(v))+on(2,3,4,u=0); we construct the terms for the 1d matrix
• Using A=mat(Vh,Vh,tgv=1e30); store these terms on a sparse matrix
• Using varf rhs(u,v)= int1d(Th,1)(f*v)+on(2,3,4,u=0); we construct the terms for the right hand side
• Using d=rhs(0,Vh,tgv=1e30); store these terms on a array larger than needed
• Using M=A(zoi,zoi); we extract the correct entries for the stiff matrix to the 1d problem and with b=d(zoi); we extract the values of right hand side. Then, our problem is Mx=b

Take a look at the following examples:

1d FreeFem++ examples

1. Steady convection-diffusion equation with remeshing example1
2. Steady reaction-diffusion equation with remeshing example2
3. Non steady heat equation equation example3
4. Non steady Black-Scholes equation example4
5. Wave equation with Crank-Nicolson example5

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