Description

EX_LAPLACE2 2D Laplace equation on a circle with nonzero boundary conditions.

[ FEA, OUT ] = EX_LAPLACE2( VARARGIN ) Laplace equation on a circle with source term 0, nonzero boundary conditions and exact solution u=r^3*sin(3*th). Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
radi        scalar {1}             Radius of circle
hmax        scalar {0.1}           Max grid cell size
sfun        string {sflag1}        Shape function
iphys       scalar 0/{1}           Use physics mode to define problem    (=1)
                                   or directly define fea.eqn/bdr fields (=0)
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { ...
   'radi',     1; ...
   'hmax',     0.1; ...
   'refsol',   'hypot(x,y)^3*sin(3*atan2(y,x))'; ...
   'sfun',     'sflag1'; ...
   'icub',     2; ...
   'iphys',    1; ...
   'iplot',    1; ...
   'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


% Geometry definition.
 gobj = gobj_circle( [0 0], opt.radi );
 fea.geom.objects = { gobj };


% Grid generation.
 fea.grid = gridgen(fea,'hmax',opt.hmax,'fid',fid,'dprim',false);
 n_bdr    = max(fea.grid.b(3,:));        % Number of boundaries.


% Problem definition.
 fea.sdim  = { 'x' 'y' };                % Coordinate names.
 if ( opt.iphys==1 )

   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} = { 0 };   % Set source term to zero.
   fea.phys.poi.bdr.coef{1,end} = repmat({opt.refsol},1,n_bdr);
   fea = parsephys(fea);                 % Check and parse physics modes.

 else

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

% Define equation system.
   fea.eqn.a.form = { [2 3;2 3] };       % First row indicates test function space   (2=x-derivative + 3=y-derivative),
% second row indicates trial function space (2=x-derivative + 3=y-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 = { 0 };               % Coefficient used in right hand side.

% Define boundary conditions.
   fea.bdr.d     = cell(1,n_bdr);
  [fea.bdr.d{:}] = deal(opt.refsol);     % Assign reference solution to all boundaries (Dirichlet).

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

 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.


% Postprocessing.
 s_err = ['abs(',opt.refsol,'-u)'];
 if ( opt.iplot>0 )
   figure
   subplot(3,1,1)
   postplot(fea,'surfexpr','u','axequal','on')
   title('Solution u')
   subplot(3,1,2)
   postplot(fea,'surfexpr',opt.refsol,'axequal','on')
   title('Exact solution')
   subplot(3,1,3)
   postplot(fea,'surfexpr',s_err,'axequal','on')
   title('Error')
 end


% Error checking.
 if ( size(fea.grid.c,1)==4 )
   xi = [0;0];
 else
   xi = [1/3;1/3;1/3];
 end
 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: %f\n',err)
   fprintf(fid,'\n\n')
 end


 out.err  = err;
 out.pass = out.err<5e-3;
 if ( nargout==0 )
   clear fea out
 end