FEATool  v1.7.1
Finite Element Analysis Toolbox
 All Files Functions Pages
ex_poisson1.m File Reference

Description

EX_POISSON1 1D Poisson equation example.

[ FEA, OUT ] = EX_POISSON1( VARARGIN ) Poisson equation on a line with a constant source term equal to 1, homogenous boundary conditions, and exact solution (-x^2+x)/2. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
hmax        scalar {1/10}          Grid cell size
sfun        string {sflag1}        Finite element shape function
iphys       scalar 0/{1}           Use physics mode to define problem    (=1)
                                   or directly define fea.eqn/bdr fields (=0)
                                   or use core assembly functions        (<0)
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output stuct

Code listing

 cOptDef = { 'hmax',   1/10;
             'sfun',   'sflag1';
             'refsol', '(-x^2+x)/2';
             'fsrc',   '1';
             'iphys',  1;
             'icub',   2;
             'iplot',  1;
             'tol',    2e-2;
             'fid',    1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


% Grid generation.
 if( opt.hmax>0 )
   nx = round( 1/opt.hmax );
   fea.grid = linegrid( nx, 0, 1 );
 else   % Scrambled testing grid.
   fea.grid.p = [ 0 1/10 4/10 1/3 1 1-1/3 ];
   fea.grid.c = [ 1 4 2 6 3 ;
                  2 3 4 5 6 ];
   fea.grid.a = [ 0 2 1 4 3 ;
                  2 4 3 0 5 ];
   fea.grid.b = [ 1 1 1 -1 ;
                  4 2 2  1 ]';
   fea.grid.s = ones(1,5);
 end
 n_bdr = 2;   % Number of boundaries.


% Problem definition.
 fea.sdim  = { 'x' };                    % Coordinate name.
 switch opt.iphys

  case 0   % Directly define fea.eqn/bdr fields.

   fea.dvar  = { 'u' };                  % Dependent variable name.
   fea.sfun  = { opt.sfun  };            % Shape function.

% Define equation system.
   fea.eqn.a.form = { [2;2] };           % First row indicates test function space   (2=x-derivative),
% second row indicates trial function space (2=x-derivative).
   fea.eqn.a.coef = { 1 };               % Coefficient used in assembling stiffness matrix.

   fea.eqn.f.form = { 1 };               % Test function space to evaluate in right hand side (1=function values).
   fea.eqn.f.coef = { opt.fsrc };        % Coefficient used in right hand side.

% Define boundary conditions.
   if( strcmp(opt.sfun(end-1:end),'H3') )   % Prescribed derivatives at end points for Hermite elements.
     fea.bdr.d = {{ 0 0 ; 1/2 -1/2 }};
   else
     fea.bdr.d     = cell(1,n_bdr);
     [fea.bdr.d{:}] = deal(0);              % Assign zero to all boundaries (homogenous Dirichlet conditions).
   end

   fea.bdr.n     = cell(1,n_bdr);        % No Neumann boundaries ('fea.bdr.n' empty).

% Parse and solve problem.
   fea       = parseprob(fea);           % Check and parse problem struct.
   fea.sol.u = solvestat(fea,'fid',fid,'icub',opt.icub); % Call to stationary solver.

  case 1   % Use physics mode.

   fea = addphys(fea,@poisson);          % Add Poisson equation physics mode.
   fea.phys.poi.sfun = { opt.sfun };     % Set shape function.
   fea.phys.poi.eqn.coef{3,4} = { opt.fsrc };   % Set source term coefficient.
   fea.phys.poi.bdr.coef{1,end} = repmat({0},1,n_bdr);   % Set Dirichlet boundary coefficient to zero.
   fea = parsephys(fea);                 % Check and parse physics modes.
   if( strcmp(opt.sfun(end-1:end),'H3') )   % Prescribed derivatives at end points for Hermite elements.
     fea.bdr.d = {{ 0 0 ; 1/2 -1/2 }};
   end

% Parse and solve problem.
   fea       = parseprob(fea);           % Check and parse problem struct.
   fea.sol.u = solvestat(fea,'fid',fid,'icub',opt.icub); % Call to stationary solver.

  otherwise   % Use core assembly functions.

   fea.dvar  = { 'u' };                  % Dependent variable name.
   fea.sfun  = { opt.sfun  };            % Shape function.
   fea       = parseprob(fea);           % Check and parse problem struct.

% Assemble stiffness matrix.
   form  = [2;2];
   sfun  = {opt.sfun;opt.sfun};
   coefa = 1;
   sind  = 1;
   i_cub = opt.icub;

   [vRowInds,vColInds,vAvals,n_rows,n_cols] = ...
     assemblea(form,sfun,coefa,i_cub,fea.grid.p,fea.grid.c,fea.grid.a,fea.grid.s,[]);
   A = sparse(vRowInds,vColInds,vAvals,n_rows,n_cols);

% Check and compare with finite difference stencil.
   if (strcmp(opt.sfun,'sflag1'))
     h = 1/nx;
     n = nx+1;
     e = ones(n,1);
     A_ref = 1/h*spdiags([-e 2*e -e], -1:1, n, n);
     A_ref(1) = A_ref(1)/2;
     A_ref(end) = A_ref(end)/2;

     err = norm(A(:)-A_ref(:));
     if err>opt.tol
       out.err  = err;
       out.pass = -1;
       return
     end
   end

   form  = 1;
   sfun  = sfun{1};
   coeff = 1;

   f = assemblef(form,sfun,coeff,i_cub,fea.grid.p,fea.grid.c,fea.grid.a,fea.grid.s,[]);


% Check and compare with finite difference stencil.
   if (strcmp(opt.sfun,'sflag1'))
     f_ref          = coeff*h*ones(n,1);
     f_ref([1 end]) = coeff*1/2*h;

     err = norm(f-f_ref);
     if err>1e-6
       out.err  = err;
       out.pass = -2;
       return
     end
   end

% Set homogenous Dirichlet boundary conditions on first and last dof/node.
   bind = [1 nx+1];
   A = A';   %'
   A(:,bind) = 0;         % Zero out Dirichlet BC rows.
   for i=1:length(bind)   % Loop to set diagonal entry to 1.
     i_a = bind(i);
     A(i_a,i_a) = 1;
   end
   A = A';   %'
   f(bind) = 0;           % Set corresponding source term entries to Dirichlet BC values.

% Solve problem.
   fea.sol.u = A\f;

 end


% Postprocessing.
 if ( opt.iplot>0 )
   x = linspace( 0, 1, 41 );
   u = evalexpr( 'u', x, fea )';
   figure
   subplot(3,1,1)
   plot( x, u )
   axis( [0 1 0 0.2])
   grid on
   title('Solution u')
   subplot(3,1,2)
   ux = (-x.^2+x)/2;
   plot( x, ux )
   axis( [0 1 0 0.2])
   grid on
   title('Exact solution')
   subplot(3,1,3)
   plot( x, abs(ux-u) )
   title('Error')
 end


% Error checking.
 xi = [1/2; 1/2];
 s_err = ['abs(',opt.refsol,'-u)'];
 err = evalexpr0(s_err,xi,1,1:size(fea.grid.c,2),[],fea);
 ref = evalexpr0('u',xi,1,1:size(fea.grid.c,2),[],fea);
 err = sqrt(sum(err.^2)/sum(ref.^2));

 if( ~isempty(fid) )
   fprintf(fid,'\nL2 Error: %e\n',err)
   fprintf(fid,'\n\n')
 end

 out.err  = err;
 out.tol  = opt.tol;
 out.pass = out.err<out.tol;
 if ( nargout==0 )
   clear fea out
 end