# Bim package

This is a short example on how to use `bim` to solve a DAR problem.

The data for this example can be found in the doc directory inside the
bim installation directory.

We want to solve the equation

** Create the mesh and precompute the mesh properties **

The geometry of the domain was created using gmsh and is stored in the file `fiume.geo`
created with gmsh

[mesh] = msh2m_gmsh("fiume","scale",1,"clscale",.1); [mesh] = bim2c_mesh_properties(mesh);

to see the mesh you can use functions from the [fpl] package

pdemesh (mesh.p, mesh.e, mesh.t) view (2)

** Construct an initial guess**

For a linear problem only the values at boundary nodes are actually relevant

xu = mesh.p(1,:).'; yu = mesh.p(2,:).'; nelems = columns(mesh.t); nnodes = columns(mesh.p); uin = 3*xu;

** Set the coefficients for the problem:**

epsilon = .1; alfa = ones(nelems,1); gamma = ones(nnodes,1); eta = epsilon*ones(nnodes,1); beta = xu+yu; delta = ones(nelems,1); zeta = ones(nnodes,1); f = ones(nelems,1); g = ones(nnodes,1);

** Construct the discretized operators**

AdvDiff = bim2a_advection_diffusion(mesh,alfa,gamma,eta,beta); Mass = bim2a_reaction(mesh,delta,zeta); b = bim2a_rhs(mesh,f,g); A = AdvDiff + Mass;

** To Apply Boundary Conditions, partition LHS and RHS**

The tags of the sides are assigned by gmsh

Dlist = bim2c_unknowns_on_side(mesh, [8 18]); ## DIRICHLET NODES LIST Nlist = bim2c_unknowns_on_side(mesh, [23 24]); ## NEUMANN NODES LIST Nlist = setdiff(Nlist,Dlist); Fn = zeros(length(Nlist),1); ## PRESCRIBED NEUMANN FLUXES Ilist = setdiff(1:length(uin),union(Dlist,Nlist)); ## INTERNAL NODES LIST

Add = A(Dlist,Dlist); Adn = A(Dlist,Nlist); ## shoud be all zeros hopefully!! Adi = A(Dlist,Ilist); And = A(Nlist,Dlist); ## shoud be all zeros hopefully!! Ann = A(Nlist,Nlist); Ani = A(Nlist,Ilist); Aid = A(Ilist,Dlist); Ain = A(Ilist,Nlist); Aii = A(Ilist,Ilist); bd = b(Dlist); bn = b(Nlist); bi = b(Ilist); ud = uin(Dlist); un = uin(Nlist); ui = uin(Ilist);

** Solve for the displacements**

temp = [Ann Ani ; Ain Aii ] \ [ Fn+bn-And*ud ; bi-Aid*ud]; un = temp(1:length(un)); ui = temp(length(un)+1:end); u(Dlist) = ud; u(Ilist) = ui; u(Nlist) = un;

** Compute the fluxes through Dirichlet sides**

Fd = Add * ud + Adi * ui + Adn*un - bd;

** Compute the gradient of the solution **

[gx, gy] = bim2c_pde_gradient(mesh,u);

** Compute the internal Advection-Diffusion flux**

[jxglob,jyglob] = bim2c_global_flux(mesh,u,alfa,gamma,eta,beta);

** Save data for later visualization**

fpl_dx_write_field("dxdata",mesh,[gx; gy]',"Gradient",1,2,1); fpl_vtk_write_field ("vtkdata", mesh, {}, {[gx; gy]', "Gradient"}, 1);