NORMAL2_2D Computes L2-norm in 2D
[err_l2]=normal2_2d(fdq,nq,errtype,x,wx,xx,jacx,y,wy,yy,jacy,...
xy,ww,nov,un,u);
Input : fdq = 0 uses Legendre Gauss quadrature formulas with nq nodes
in each element (exactness degree = 2*nq+1)
= 1 uses Legendre Gauss Lobatto quadrature formulas with npdx/y
quadrature nodes in each element.
Quadrature nodes are the nodes of the mesh.
nq = nodes (in each element and along each direction)
for GL quadrature formulas. Not used if fdq == 1
errtype = 0 for absolute error ||u-u_ex||
1 for relative error ||u-u_ex||/||u_ex||
x = column array with npdx LGL nodes in [-1,1]
wx= column array with npdx LGL weigths in [-1,1]
dx= derivative matrix produced with derlgl
xx = 2-indexes array of size (4,ne)
xx(1:4,ie)=[x_V1_ie;x_V2_ie;x_V3_ie;x_V4_ie]
(ne=nex*ney is the global number of spectral elements)
jacx = array (length(jacx)=ne); jacx(ie)= (x_V2_ie-x_V1_ie)/2
y = npdy LGL nodes in [-1,1], previously generated by xwlgl
wy = npdy LGL weigths in [-1,1], previously generated by xwlgl
dy= derivative matrix produced with derlgl
yy = 2-indexes array of size (4,ne):
yy(1:4,ie)=[y_V1_ie;y_V2_ie;y_V3_ie;y_V4_ie]
jacy = array (length(jacy)=ne); jacy(ie)= (y_V3_ie-y_V1_ie)/2
xy = column array with global mesh, length: noe=nov(npdx*npdy,ne)
ww = column array with global weigths, length: noe=nov(npdx*npdy,ne)
diag(ww) is the mass matrix associated to SEM discretization
nov = local to global map. 2-indexes array, size(nov)=[nov,ne]
un = column array with the numerical solution
u = column array with the evaluation of exact solution
uex = exact solution (uex=@(x,y)[uex(x,y)], with .*, .^, ./)
Output: err_l2 = error in L2-norm
Reference: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
"Spectral Methods. Fundamentals in Single Domains"
Springer Verlag, Berlin Heidelberg New York, 2006.0001 function [err_l2]=normal2_2d(fdq,nq,errtype,x,wx,xx,jacx,y,wy,yy,jacy,... 0002 xy,ww,nov,un,u,uex); 0003 % NORMAL2_2D Computes L2-norm in 2D 0004 % 0005 % [err_l2]=normal2_2d(fdq,nq,errtype,x,wx,xx,jacx,y,wy,yy,jacy,... 0006 % xy,ww,nov,un,u); 0007 % 0008 % Input : fdq = 0 uses Legendre Gauss quadrature formulas with nq nodes 0009 % in each element (exactness degree = 2*nq+1) 0010 % = 1 uses Legendre Gauss Lobatto quadrature formulas with npdx/y 0011 % quadrature nodes in each element. 0012 % Quadrature nodes are the nodes of the mesh. 0013 % nq = nodes (in each element and along each direction) 0014 % for GL quadrature formulas. Not used if fdq == 1 0015 % errtype = 0 for absolute error ||u-u_ex|| 0016 % 1 for relative error ||u-u_ex||/||u_ex|| 0017 % x = column array with npdx LGL nodes in [-1,1] 0018 % wx= column array with npdx LGL weigths in [-1,1] 0019 % dx= derivative matrix produced with derlgl 0020 % xx = 2-indexes array of size (4,ne) 0021 % xx(1:4,ie)=[x_V1_ie;x_V2_ie;x_V3_ie;x_V4_ie] 0022 % (ne=nex*ney is the global number of spectral elements) 0023 % jacx = array (length(jacx)=ne); jacx(ie)= (x_V2_ie-x_V1_ie)/2 0024 % y = npdy LGL nodes in [-1,1], previously generated by xwlgl 0025 % wy = npdy LGL weigths in [-1,1], previously generated by xwlgl 0026 % dy= derivative matrix produced with derlgl 0027 % yy = 2-indexes array of size (4,ne): 0028 % yy(1:4,ie)=[y_V1_ie;y_V2_ie;y_V3_ie;y_V4_ie] 0029 % jacy = array (length(jacy)=ne); jacy(ie)= (y_V3_ie-y_V1_ie)/2 0030 % xy = column array with global mesh, length: noe=nov(npdx*npdy,ne) 0031 % ww = column array with global weigths, length: noe=nov(npdx*npdy,ne) 0032 % diag(ww) is the mass matrix associated to SEM discretization 0033 % nov = local to global map. 2-indexes array, size(nov)=[nov,ne] 0034 % un = column array with the numerical solution 0035 % u = column array with the evaluation of exact solution 0036 % uex = exact solution (uex=@(x,y)[uex(x,y)], with .*, .^, ./) 0037 % 0038 % Output: err_l2 = error in L2-norm 0039 % 0040 % Reference: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, 0041 % "Spectral Methods. Fundamentals in Single Domains" 0042 % Springer Verlag, Berlin Heidelberg New York, 2006. 0043 0044 % Written by Paola Gervasio 0045 % $Date: 2007/04/01$ 0046 0047 0048 0049 npdx=length(x); npdy=length(y); 0050 [ldnov,ne]=size(nov); noe=nov(ldnov,ne); 0051 num=0; den=0; 0052 err_l2=0; 0053 0054 if fdq==0 0055 0056 % Legendre Gauss nodes and weigths 0057 0058 [xg,wxg] = xwlg(nq,-1,1); 0059 [yg,wyg] = xwlg(nq,-1,1); 0060 [wxg1,wyg1]=meshgrid(wxg,wyg); wxyg=wxg1.*wyg1;wxyg=wxyg';wxyg=wxyg(:); 0061 clear wxg1 wyg1; 0062 0063 % Evaluation of Lagrange basis polynomials at quadrature nodes xg 0064 % 0065 [phix]= intlag_lgl (x,xg); 0066 [phiy]= intlag_lgl (y,yg); 0067 0068 % Loop on spectral elements 0069 0070 for ie=1:ne 0071 un_loc=un(nov(1:ldnov,ie)); 0072 [norma_loc1,norma_loc2]=normal2_ex_loc(errtype,nq,uex,un_loc,wxyg,... 0073 jacx(ie),xx(1:4,ie),xg,phix,... 0074 jacy(ie),yy(1:4,ie),yg,phiy); 0075 num=num+norma_loc1; 0076 den=den+norma_loc2; 0077 end 0078 0079 elseif fdq==1 0080 0081 num=sum((u-un).^2.*ww); 0082 den=sum(u.^2.*ww); 0083 0084 end 0085 0086 0087 if errtype==0 0088 err_l2=sqrt(num); 0089 elseif errtype==1 0090 if abs(den)>1.d-14; err_l2=sqrt(num/den); end 0091 end 0092 0093 return 0094 0095 function [norma_loc1,norma_loc2]=normal2_ex_loc(errtype,nq,uex,... 0096 un,wxyg,jacx,xx,xg,phix,... 0097 jacy,yy,yg,phiy); 0098 0099 % High degree Legendre Gaussian formulas to compute H1-norm error 0100 0101 [nq,npdx]=size(phix); 0102 [nq,npdy]=size(phiy); 0103 0104 % mapping quadrature nodes on element ie 0105 0106 xxg=xg*jacx+(xx(2)+xx(1))*.5; 0107 yyg=yg*jacy+(yy(3)+yy(1))*.5; 0108 [xg1,yg1]=meshgrid(xxg,yyg); xg1=xg1'; yg1=yg1'; xyg=[xg1(:),yg1(:)]; 0109 clear xg1; clear yg1; 0110 0111 0112 % evaluation of exact solution at quadrature nodes. 0113 0114 [U]=uex(xyg(:,1),xyg(:,2)); 0115 0116 0117 % evaluate numerical solution at quadrature nodes. 0118 0119 un=reshape(un,npdx,npdy); 0120 0121 un_i=phix*(phiy*un')'; 0122 0123 un_i=un_i(:); 0124 0125 0126 % compute the sum 0127 0128 norma_loc1=sum((U-un_i).^2.*wxyg)*jacx*jacy; 0129 0130 if errtype==0 0131 norma_loc2=0; 0132 else 0133 norma_loc2=sum(U.^2.*wxyg)*jacx*jacy; 0134 end 0135 0136 return 0137 0138