FEATool Multiphysics  v1.14 Finite Element Analysis Toolbox
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 struct


# 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.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