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QDDGOXnlpoisson.m

function [V,n,p,res,niter] = QDDGOXnlpoisson (mesh,Dsides,Sinodes,SiDnodes,...
                                              Sielements,Vin,nin,pin,...
                                              Fnin,Fpin,Gin,Gpin,D,l2,l2ox,...
                                              toll,maxit,verbose)

%  
%   [V,n,p,res,niter] = QDDGOXnlpoisson (mesh,Dsides,Sinodes,SiDnodes,...
%     Sielements,Vin,nin,pin,...
%     Fnin,Fpin,Gin,Gpin,D,l2,l2ox,...
%     toll,maxit,verbose)
%
%  solves $$ -\lambda^2 V'' + (n(V,Fn) - p(V,Fp) -D)$$
%

global DDGOXNLPOISSON_LAP DDGOXNLPOISSON_MASS DDGOXNLPOISSON_RHS 

%% Set some useful constants
dampit            = 3;
dampcoeff   = 2;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% convert input vectors to columns
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if Ucolumns(D)>Urows(D)
      D=D';
end


if Ucolumns(nin)>Urows(nin)
      nin=nin';
end

if Ucolumns(pin)>Urows(pin)
      pin=pin';
end

if Ucolumns(Vin)>Urows(Vin)
      Vin=Vin';
end 

if Ucolumns(Fnin)>Urows(Fnin)
      Fnin=Fnin';
end 

if Ucolumns(Fpin)>Urows(Fpin)
      Fpin=Fpin';
end 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% setup FEM data structures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

nodes=mesh.p;
elements=mesh.t;
Nnodes = length(nodes);
Nelements = length(elements);

Dedges    =[];

for ii = 1:length(Dsides)
      Dedges=[Dedges,find(mesh.e(5,:)==Dsides(ii))];
end

% Set list of nodes with Dirichelet BCs
Dnodes = mesh.e(1:2,Dedges);
Dnodes = [Dnodes(1,:) Dnodes(2,:)];
Dnodes = unique(Dnodes);

% Set values of Dirichelet BCs
Bc     = zeros(length(Dnodes),1);
% Set list of nodes without Dirichelet BCs
Varnodes = setdiff([1:Nnodes],Dnodes);


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%          initialization:
%%          we're going to solve
%%          $$ - \lambda^2 (\delta V)'' +  (\frac{\partial n}{\partial V} - \frac{\partial p}{\partial V})= -R $$
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% set $$ n_1 = nin $$ and $$ V = Vin $$
V = Vin;
Fn = Fnin;
Fp = Fpin;
G  = Gin;
Gp = Gpin;
n = exp(V(Sinodes)+G-Fn);
p = exp(-V(Sinodes)-Gp+Fp);
n(SiDnodes) = nin(SiDnodes);
p(SiDnodes) = pin(SiDnodes);


%%%
%%% Compute LHS matrices
%%%

%% let's compute  FEM approximation of $$ L = -  \frac{d^2}{x^2} $$
if (isempty(DDGOXNLPOISSON_LAP))
      coeff = l2ox * ones(Nelements,1);
      coeff(Sielements)=l2;
      DDGOXNLPOISSON_LAP = Ucomplap (mesh,coeff);
end

%% compute $$ Mv = ( n + p)  $$
%% and the (lumped) mass matrix M
if (isempty(DDGOXNLPOISSON_MASS))
    Cvect = zeros(Nelements,1);
    Cvect(Sielements)=1;
      DDGOXNLPOISSON_MASS = Ucompmass2 (mesh,ones(Nnodes,1),Cvect);
end
freecarr=zeros(Nnodes,1);
freecarr(Sinodes)=(n + p);
Mv      =  freecarr;
MV(SiDnodes) = 0;
M       =  DDGOXNLPOISSON_MASS*sparse(diag(Mv));

%%%
%%% Compute RHS vector (-residual)
%%%

%% now compute $$ T0 = \frac{q}{\epsilon} (n - p - D) $$
if (isempty(DDGOXNLPOISSON_RHS))
      DDGOXNLPOISSON_RHS = Ucompconst (mesh,ones(Nnodes,1),ones(Nelements,1));
end
totcharge = zeros(Nnodes,1);
totcharge(Sinodes)=(n - p - D);
Tv0   = totcharge;
T0    = Tv0 .* DDGOXNLPOISSON_RHS;

%% now we're ready to build LHS matrix and RHS of the linear system for 1st Newton step
A           = DDGOXNLPOISSON_LAP + M;
R           = DDGOXNLPOISSON_LAP * V  + T0; 

%% Apply boundary conditions
A (Dnodes,:) = [];
A (:,Dnodes) = [];
R(Dnodes)  = [];

%% we need $$ \norm{R_1} $$ and $$ \norm{R_k} $$ for the convergence test   
normr(1)    =  norm(R,inf);
relresnorm  = 1;
reldVnorm       = 1;
normrnew    = normr(1);
dV              = V*0;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% START OF THE NEWTON CYCLE
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for newtit=1:maxit
      if (verbose>0)
            fprintf(1,'\n***\nNewton iteration: %d, reldVnorm = %e\n***\n',newtit,reldVnorm);
            
      end

      dV(Varnodes) =(A)\(-R);
      dV(Dnodes)=0;
      
      
      %%%%%%%%%%%%%%%%%%
      %% Start of th damping procedure
      %%%%%%%%%%%%%%%%%%
      tk = 1;
      for dit = 1:dampit
            if (verbose>0)
                  fprintf(1,'\ndamping iteration: %d, residual norm = %e\n',dit,normrnew);
            end
            Vnew   = V + tk * dV;
            
            n = exp(Vnew(Sinodes)+G-Fn);
            p = exp(-Vnew(Sinodes)-Gp+Fp);
            n(SiDnodes) = nin(SiDnodes);
            p(SiDnodes) = pin(SiDnodes);
            
            
            %%%
            %%% Compute LHS matrices
            %%%
            
            %% let's compute  FEM approximation of $$ L = -  \frac{d^2}{x^2} $$
            %L      = Ucomplap (mesh,ones(Nelements,1));
            
            %% compute $$ Mv =  ( n + p)  $$
            %% and the (lumped) mass matrix M
            freecarr=zeros(Nnodes,1);
            freecarr(Sinodes)=(n + p);  
            Mv   =  freecarr;
            M       =  DDGOXNLPOISSON_MASS*sparse(diag(Mv));%M     = Ucompmass (mesh,Mv);
            
            %%%
            %%% Compute RHS vector (-residual)
            %%%
            
            %% now compute $$ T0 = \frac{q}{\epsilon} (n - p - D) $$
            totcharge( Sinodes)=(n - p - D);
            Tv0   = totcharge;
            T0    = Tv0 .* DDGOXNLPOISSON_RHS;%T0    = Ucompconst (mesh,Tv0,ones(Nelements,1));
            
            %% now we're ready to build LHS matrix and RHS of the linear system for 1st Newton step
            A           = DDGOXNLPOISSON_LAP + M;
            R           = DDGOXNLPOISSON_LAP * Vnew  + T0; 
            
            %% Apply boundary conditions
            A (Dnodes,:) = [];
            A (:,Dnodes) = [];
            R(Dnodes)  = [];
            
            %% compute $$ | R_{k+1} | $$ for the convergence test
            normrnew= norm(R,inf);
            
            % check if more damping is needed
            if (normrnew > normr(newtit))
                  tk = tk/dampcoeff;
            else
                  if (verbose>0)
                          fprintf(1,'\nexiting damping cycle because residual norm = %e \n-----------\n',normrnew);
                  end         
                  break
            end   
      end
        
      V                       = Vnew;     
      normr(newtit+1)   = normrnew;
      dVnorm              = norm(tk*dV,inf);
      pause(.1);
        % check if convergence has been reached
      reldVnorm           = dVnorm / norm(V,inf);
      if (reldVnorm <= toll)
          if(verbose>0)
                  fprintf(1,'\nexiting newton cycle because reldVnorm= %e \n',reldVnorm);
          end
          break
      end
        
end

res = normr;
niter = newtit;




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