# Bim package

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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.

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

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


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:

${\displaystyle -div(\alpha \gamma (\eta \nabla u-\beta u))+\delta \zeta u=fg}$

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);


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!!

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);


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

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