README.md

-# PDE-VKOGA-paper-experiments
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README.txt

+This directory provides the code for the experiments
+to the paper
+"Adaptive meshfree approximation for linear
+elliptic partial differential equations with
+PDE-greedy kernel methods" by
+Tizian Wenzel, Daniel Winkle, Gabriele Santin, and Bernard Haasdonk
+
+More specific:
+
+For reproducing the FEM convergence table on the sector example
+of Section 6.1, please
+
+a) Install RBmatlab from the website www.morepas.org/software
+
+The code can be run if RBmatlab from version 16.09 is installed
+
+b) run the script fem_sector_example.m from this directory
+
+Caution:
+
+This operation is very expensive (takes several hours) as hundred of
+thousands of point with global coordinates need to be searched and found
+in FEM meshes. This is required to be consistent in the error computation
+to other approximation techniques using these uniform test grids.
+
+
+
+

fem_sector_example.m

+function fem_sector_example(step)
+%function fem_sector_example(step)
+%
+% Demonstration of FEM error convergence for sector example.
+%
+% This operation is very expensive (takes several hours) as hundred of
+% thousands of point with global coordinates need to be searched and found
+% in FEM meshes. This is required to be consistent in the error computation
+% to other approximation techniques using these uniform test grids.
+%
+% The code can be run if RBmatlab from version 16.09 is installed
+% as obtained from www.morepas.org/software 
+
+% B. Haasdonk 8.5.2024
+
+if nargin < 1
+  step = 4;
+end;
+
+disp('FEM error convergence study for sector geometry')
+disp('Note that the global to local coordinate search is extremely')
+disp('expensive due to the many test points, and will overall take')
+disp('several hours.')
+
+switch step
+  
+%  case 1 %  create scale of refined gridfiles
+%    fns = {'sectorg_alpha5over3','sectorg_alpha2over3'};
+%    for fni = 1:length(fns)
+%      fn = fns{fni}
+%      [p,e,t] = initmesh(fn);
+%      meshfile = [fn,'_r0.mat'];
+%      save(meshfile,'p','e','t');
+%      for i = 1:4
+%	[p,e,t] = refinemesh(fn,p,e,t);
+%	meshfile = [fn,'_r',num2str(i),'.mat'];
+%	save(meshfile,'p','e','t');  
+%      end;
+%    end;
+%    disp('mesh sequence generated and stored');
+
+%    % continue with error computation: 
+%    fem_sector_example(4);
+    
+ case 2 % load and plot mesh
+
+  load('sectorg_alpha5over3_r1','p','t');
+  grid = triagrid(p,t,[]);
+  plot(grid);
+  axis equal;
+
+  figure;
+  load('sectorg_alpha2over3_r1','p','t');
+  grid = triagrid(p,t,[]);
+  plot(grid);
+  axis equal;
+
+ case 3 % fem on mesh 
+  
+  params = [];
+%  params.solution_number = 1; % example with smooth solution 
+  params.solution_number = 2; % example with singular sol, inhom bv
+
+%  params.alpha = 5/3;
+%  params.grid_initfile = 'sectorg_alpha5over3_r1.mat';
+%  params.grid_initfile = 'sectorg_alpha5over3_r2.mat';
+%  params.grid_initfile = 'sectorg_alpha5over3_r3.mat';
+%  params.grid_initfile = 'sectorg_alpha5over3_r4.mat';
+  params.alpha = 2/3;
+  params.grid_initfile = 'sectorg_alpha2over3_r1.mat';
+%  params.grid_initfile = 'sectorg_alpha2over3_r2.mat';
+%  params.grid_initfile = 'sectorg_alpha2over3_r3.mat';
+%  params.grid_initfile = 'sectorg_alpha2over3_r4.mat';
+  model = pacman_model(params);
+  % generate grid and fem matrices:
+  model_data = gen_model_data(model);
+  figure, plot(model_data.grid);
+  axis equal; axis tight; title('FEM grid')  
+  sim_data = detailed_simulation(model, model_data); 
+  % plot results
+  figure, plot_sim_data(model,model_data,sim_data,[]);
+  title('FEM solution of -Laplace u = f')
+  axis equal;
+  axis tight;
+%  disp(['ndofs = ',num2str(sim_data.uh.df_info.ndofs)]);
+
+ case 4 % fem on mesh and convergence study 
+     
+  params = [];
+  params.solution_number = 1; % example with smooth solution 
+%  params.solution_number = 2; % example with singular sol, inhom bv
+
+    disp(' ');
+    disp('ndofs  |    L2-error    |     H1-error    |     infty-error    |   L2-error-grid  | infty-error-grid    |    t_CPU')
+    disp('---------------------------------------------------------------------------------------------------------------------')
+    
+  for i = 1:4
+    % params.alpha = 5/3;
+    % params.grid_initfile = ['sectorg_alpha5over3_r',num2str(i),'.mat'];
+    %  params.grid_initfile = 'sectorg_alpha5over3_r2.mat';
+    %  params.grid_initfile = 'sectorg_alpha5over3_r3.mat';
+    %  params.grid_initfile = 'sectorg_alpha5over3_r4.mat';
+    params.alpha = 2/3;
+    params.grid_initfile = ['sectorg_alpha2over3_r',num2str(i),'.mat'];
+    %  params.grid_initfile = 'sectorg_alpha2over3_r2.mat';
+    %  params.grid_initfile = 'sectorg_alpha2over3_r3.mat';
+    %  params.grid_initfile = 'sectorg_alpha2over3_r4.mat';
+    model = pacman_model(params);
+    % generate grid and fem matrices:
+    tic
+    model_data = gen_model_data(model);
+    sim_data = detailed_simulation(model, model_data); 
+    t = toc;
+    % error computation:  
+    % project exact solution onto higher degree polynomial fem func
+    par.pdeg = 4;
+    par.qdeg = 8;
+    par.dimrange = 1;    
+    p4_df_info = feminfo(par,model_data.grid);
+    
+    uexact_h = femdiscfunc([],p4_df_info);
+    uexact_h = fem_interpol_global(model.solution,uexact_h);
+    uh = femdiscfunc([],p4_df_info);
+    u_local_eval = @(grid,elids,lcoord,params) ...
+	my_uh_local_eval(grid,elids,lcoord,params,sim_data.uh); 
+    uh = fem_interpol_local(u_local_eval,uh);
+    
+    err = uh - uexact_h;
+    if i == 1
+      plot(err);
+      title('error for i=1')
+      axis equal;
+      axis tight;
+    end;
+%    keyboard;
+    l2err = fem_l2_norm(err);
+    h1err = fem_h1_norm(err);
+    linftyerr = max(abs(err.dofs));
+    [l2err_uniformgrid, linftyerr_uniformgrid] = errors_uniformgrid(model,uh);
+%    l2err_uniformgrid = zeros(size(l2err));
+%    linftyerr_uniformgrid = zeros(size(l2err));
+    ndofs = sim_data.uh.df_info.ndofs;
+    disp([num2str(ndofs,'%10.4d'),'   |   ',...
+	  num2str(l2err,'%10.5e'),'   |   ',...
+	  num2str(h1err,'%10.5e'),'   |   ',...
+	  num2str(linftyerr,'%10.5e'),'   |   ',...
+	  num2str(l2err_uniformgrid,'%10.5e'),'   |   ',...
+	  num2str(linftyerr_uniformgrid,'%10.5e'),'   |   ',...
+      num2str(t,'%10.5e')]);
+  end;
+
+ otherwise
+  error('step number unknown');
+end;
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%    auxiliary functions
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+function [l2err_uniformgrid, linftyerr_uniformgrid] = errors_uniformgrid(model, uh);  
+% determine test points
+  h = 0.002;
+%  h = 0.002;
+%  h = 0.002;
+  x = -1:h:1;
+  [XX,YY] = meshgrid(x,x);  
+  i = find((XX.^2+YY.^2)<=1);
+  XX = XX(i);
+  YY = YY(i);
+  Phi = atan(YY./XX);
+  i = find(Phi<0 & XX>=0);
+  Phi(i) = Phi(i) + 2*pi;
+  i = find(XX<0);
+  Phi(i) = Phi(i) + pi;
+  i = find(Phi<model.alpha*pi);
+  XX = XX(i);
+  YY = YY(i);
+  Phi = Phi(i);
+  %scatter3(XX,YY,Phi)
+  test_sol = model.solution([XX,YY]);
+  test_uh = zeros(size(test_sol));  
+  % evaluate fem approximation in global points
+  % bad... : loop over points, should be vectorized...
+  f = waitbar(0,'Iterating over points');
+  for i = 1:length(Phi);
+    if mod(i,1000)==0;
+      waitbar(i/length(Phi),f,'Iterating over points');
+    end;
+    eind = find_triangle(uh.grid,[XX(i),YY(i)]);
+    if eind>0
+      lcoord = global2local(uh.grid,eind,[XX(i),YY(i)]);
+      test_uh(i) = fem_evaluate(uh,eind,lcoord);
+      % sanity check: transform local coordinate back to global:
+      p = local2global(uh.grid,eind,lcoord,[]);
+      d = p-[XX(i),YY(i)];
+%      if norm(d)>10*eps
+%	disp('norm of reconstructed point too large!, please inspect');%
+%	keyboard
+%      end;     
+    else
+      test_uh(i) = NaN;
+    end;
+  end;
+  close(f);
+  
+  linftyerr_uniformgrid = max(abs(test_sol-test_uh));
+  i = find(~isnan(test_uh));
+  l2err_uniformgrid = sqrt(h*h*sum((test_sol(i)-test_uh(i)).^2));
+
+  %figure;scatter3(XX,YY,double(isnan(test_uh)))
+%  if isnan(l2err_uniformgrid)
+%    disp('NaN in error!')
+%    keyboard;
+%  end;
+  
+function eind = find_triangle(grid,glob)
+% returns the index of a triangle in grid containing the global point glob
+% if no triangle is found, then -1 is returned.
+inside = ones(grid.nelements,1);
+for j = 1:3 % check if point is "above" edge connecting point j to j+1
+  jp1 = mod(j,3)+1;
+  jp2 = mod(jp1,3)+1;
+  Xj = grid.X(grid.VI(:,j));
+  Yj = grid.Y(grid.VI(:,j));
+  Xjp1 = grid.X(grid.VI(:,jp1));
+  Yjp1 = grid.Y(grid.VI(:,jp1));
+  Vjjp1 = [Xjp1-Xj, Yjp1-Yj];
+  Vjglob = [glob(1)*ones(size(Xj)) - Xj, glob(2)*ones(size(Yj)) - Yj];
+  crossz = sign(Vjjp1(:,1).*Vjglob(:,2) - Vjjp1(:,2).*Vjglob(:,1));
+  if j==1
+    Xjp2 = grid.X(grid.VI(:,jp2));
+    Yjp2 = grid.Y(grid.VI(:,jp2));
+    Vjjp2 = [Xjp2-Xj, Yjp2-Yj];
+    crossz2 = sign(Vjjp1(:,1).*Vjjp2(:,2) - Vjjp1(:,2).*Vjjp2(:,1)); % orientation of trias
+  end;
+  i = find(crossz.*crossz2<0);
+  inside(i) = 0;
+end;
+eind = find(inside);
+if isempty(eind)
+  eind = -1;
+end
+if length(eind)>1
+  %  disp('length eind > 1, please check! ');
+  eind = eind(1);
+end;
+%if length(eind)==1
+%  disp('length eind == 1, nice :-)  ');
+%end;
+
+% settings for pacman model
+function model = pacman_model(params);
+  if ~isfield(params,'alpha')
+    alpha = 5/3;
+  else
+    alpha = params.alpha;
+  end;
+%  disp(['chosen alpha = ',num2str(alpha)]);
+  %alpha = 5/3; % hard coded in sectorg.m if changed here, change there!
+  model = poisson_model(params);
+  model.alpha = alpha;
+  model = rmfield(model,{'boundary_type','normals',...
+			 'xnumintervals','ynumintervals','xrange','yrange'});
+  model.has_reaction = 0;
+  model.has_advection = 0;
+  model.has_diffusivity = 1;
+  model.has_source = 1;
+  model.has_dirichlet_values = 1;
+  model.has_neumann_values = 0;
+  model.has_robin_values = 0;
+  model.compute_output_functional = 0;
+  switch params.solution_number
+    case 1 % smooth solution
+      model.solution = @(glob,params) ...
+			sum(glob.^2,2);
+      model.source = @(glob,params) ...
+		     - 4 * ones(size(glob,1),1);
+    case 2 % solution with singularity and inhomogeneous bnd val.
+      model.solution = @(glob,params) ...
+			pacman_exact_solution(glob',alpha);
+      model.source = @(glob,params) ...
+		      neg_Laplace_pacman_exact_solution(glob',alpha);
+      % case 3 % solution wiht singularity and homogeneous bnd val.
+      %  % someting seems to be buggy here, no convergence observed...
+      %  model.solution = @(glob,params) ...
+      %      pacman_exact_solution2(glob',alpha);
+      %  model.source = @(glob,params) ...
+      %      neg_Laplace_pacman_exact_solution2(glob',alpha);
+      %  error('please use solution_number=1 or 2.')
+  end;
+  model.diffusivity_tensor = @(glob,params) ...
+			      [ones(size(glob,1),1),...
+			       zeros(size(glob,1),1),...
+			       zeros(size(glob,1),1),...
+			       ones(size(glob,1),1)];
+  model.reaction = @(glob,params) zeros(size(glob,1),1);
+  model.dirichlet_values = @(glob,params) ...
+			    params.solution(glob,params);
+  model.grid_initfile = params.grid_initfile;
+  model.gridtype = 'triagrid';
+  model.pdeg = 1;
+  model.qdeg = 2;
+  model.dimrange = 1;
+  model = elliptic_discrete_model(model);
+	     %model.detailed_simulation = @pacman_detailed_simulation;
+
+function f = pacman_exact_solution(x,alpha);
+% function u(x) = |x|^(1/alpha)sin(phi(x)/alpha)
+% with phi(x) = atan(x2/x1).
+% which has -Laplace u = 0 on pacman shape
+% but non-homogeneous boundary values.
+f1 = sum(x.^2,1).^(0.5/alpha);
+phi = atan(x(2,:)./x(1,:));
+i = find(phi<0 & x(1,:)>=0);
+phi(i) = phi(i) + 2*pi;
+i = find(x(1,:)<0);
+phi(i) = phi(i) + pi;
+f2 = sin(phi/alpha);
+i = find(isnan(f2));
+f2(i) = 0;
+f = f1.*f2;
+
+function f = neg_Laplace_pacman_exact_solution(x,alpha);
+f = zeros(size(x,2),1);
+
+function res = my_uh_local_eval(grid,elids,lcoord,params,df)
+% dummy function used for evaluating a discrete function at finer
+% lagrange-grid nodes
+res = fem_evaluate(df,elids,lcoord,[],[]);

sectorg_alpha2over3.m

+function [x,y]=sectorg_alpha2over3(bs,s)
+
+nbs=3;
+alpha = 2/3 * pi;
+
+if nargin==0,
+  x=nbs; % number of boundary segments
+  return
+end
+
+d=[
+  0  0 1% start parameter value
+  1  alpha 0% end parameter value
+  1  1 1% left hand region
+  0  0 0% right hand region
+];
+
+bs1=bs(:)';
+
+if find(bs1<1 | bs1>nbs),
+  error('semicircleg:InvalidBs', 'Non existent boundary segment number.')
+end
+
+if nargin==1,
+  x=d(:,bs1);
+  return
+end
+
+x=zeros(size(s));
+y=zeros(size(s));
+[m,n]=size(bs);
+if m==1 && n==1,
+  bs=bs*ones(size(s)); % expand bs
+elseif m~=size(s,1) || n~=size(s,2),
+  error('semicircleg:SizeBs', 'bs must be scalar or of same size as s.');
+end
+
+if ~isempty(s),
+  % boundary segment 1
+  ii=find(bs==1);
+  x(ii) = s(ii);
+  y(ii) = zeros(size(ii));
+
+  % boundary segment 2
+  ii=find(bs==2);
+  x(ii) = cos(s(ii));
+  y(ii) = sin(s(ii));
+  
+  
+  % boundary segment 3
+  ii=find(bs==3);
+  x(ii) = s(ii)*cos(alpha);
+  y(ii) = s(ii)*sin(alpha);
+end
+