ERRORS_1D Computes errors for 1D b.v.p.
Input:
nx = polynomial degree in each element (the same in each element)
to be set only if p=4, otherwise, nx=p;
ne = number of elements (equally spaced)
xa, xb = extrema of computational domain Omega=(xa,xb)
un = numerical solution produced, e.g., by lap_1d, ellprecofem_1d,..
uex = exact solution (uex=@(x)[uex(x)], with .*, .^, ./)
uexx = first derivative of exact solution (uexx=@(x)[uexx(x)],
with .*, .^, ./)
param(1) = 1: compute errors (L^inf-norm, L2-norm, H1-norm)
on the exact solution
2: no errors are computed
param(2) = 0: LG quadrature formulas with high precision degree are
used to compute norms (exact norms)
1: LGL quadrature formulas with npdx,npdy nodes are
used to compute norms (discrete norms)
(used only if param(1) == 1)
param(3) = number of nodes for high degree quadrature formula,
(used only if param(2) == 0 & param(1) == 1)
param(4) = 0: absolute errors are computed
1: relative errors are computed
(used only if param(1) == 1)
Output:
err_inf = ||u_ex-u_N|| with respect to discrete maximum-norm
err_h1 = ||u_ex-u_N|| with respect to discrete H1-norm
err_l2 = ||u_ex-u_N|| with respect to discrete 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_inf,err_h1,err_l2]=errors_1d(nx,ne,xa,xb,un,uex,uexx,param); 0002 % ERRORS_1D Computes errors for 1D b.v.p. 0003 % 0004 % Input: 0005 % nx = polynomial degree in each element (the same in each element) 0006 % to be set only if p=4, otherwise, nx=p; 0007 % ne = number of elements (equally spaced) 0008 % xa, xb = extrema of computational domain Omega=(xa,xb) 0009 % un = numerical solution produced, e.g., by lap_1d, ellprecofem_1d,.. 0010 % uex = exact solution (uex=@(x)[uex(x)], with .*, .^, ./) 0011 % uexx = first derivative of exact solution (uexx=@(x)[uexx(x)], 0012 % with .*, .^, ./) 0013 % param(1) = 1: compute errors (L^inf-norm, L2-norm, H1-norm) 0014 % on the exact solution 0015 % 2: no errors are computed 0016 % param(2) = 0: LG quadrature formulas with high precision degree are 0017 % used to compute norms (exact norms) 0018 % 1: LGL quadrature formulas with npdx,npdy nodes are 0019 % used to compute norms (discrete norms) 0020 % (used only if param(1) == 1) 0021 % param(3) = number of nodes for high degree quadrature formula, 0022 % (used only if param(2) == 0 & param(1) == 1) 0023 % param(4) = 0: absolute errors are computed 0024 % 1: relative errors are computed 0025 % (used only if param(1) == 1) 0026 % 0027 % Output: 0028 % err_inf = ||u_ex-u_N|| with respect to discrete maximum-norm 0029 % err_h1 = ||u_ex-u_N|| with respect to discrete H1-norm 0030 % err_l2 = ||u_ex-u_N|| with respect to discrete L2-norm 0031 % 0032 % Reference: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, 0033 % "Spectral Methods. Fundamentals in Single Domains" 0034 % Springer Verlag, Berlin Heidelberg New York, 2006. 0035 0036 % Written by Paola Gervasio 0037 % $Date: 2007/04/01$ 0038 0039 npdx=nx+1; 0040 0041 [x,wx]=xwlgl(npdx); dx=derlgl(x,npdx); 0042 0043 % nov 0044 ldnov=npdx; nov=zeros(ldnov,ne); 0045 [nov]=cosnov_1d(npdx,ne,nov); 0046 noe=nov(npdx,ne); 0047 0048 % Uniform decomposition of Omega in ne elements 0049 [xx,jacx,xy,ww]=mesh_1d(xa,xb,ne,npdx,nov,x,wx); 0050 0051 if(param(1)==1) 0052 % Evaluate exact solution to compute errors 0053 nq=param(3); fdq=param(2); err_type=param(4); 0054 0055 u=uex(xy); 0056 0057 % 0058 0059 err=u-un; 0060 0061 % ||u_ex-u_N|| with respect to discrete maximum-norm 0062 0063 if err_type==0 0064 err_inf=norm(err,inf); 0065 else 0066 err_inf=norm(err,inf)/norm(u,inf); 0067 end 0068 0069 0070 % ||u_ex-u_N|| with respect to H1 norm 0071 % fdq=0 LG quadrature formula with nq nodes in each element 0072 % fdq=1 LGL quadrature formula with npdx nodes in each element 0073 [err_h1]=normah1_1d(fdq, nq, err_type, un, uex, uexx,... 0074 x, wx, dx, xx, jacx, xy,nov); 0075 0076 % ||u_ex-u_N|| with respect to L2 norm 0077 0078 [err_l2]=normal2_1d(fdq, nq, err_type, un, uex, ... 0079 x, wx, xx, jacx, xy,nov); 0080 end 0081 0082 return