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{"@context":"https://schema.org","@type":"DiscussionForumPosting","headline":"The missing sqrt(2) of some basis in Degree 2 Arnold–Winther on a triangle","articleBody":"https://defelement.org/elements/examples/triangle-arnold-winther-2.html\n\nI found that the dual basis \\Phi_17, \\Phi_18, \\Phi_19, \\Phi_20 should be multiplied by sqrt(2) (although several hours ago, they were \\Phi_9 to \\Phi_12), the others are correct. For example, l_17(\\Phi_17)= sqrt(2)/2, rather than 1. This suggests the sqrt(2) is missing. The following is the matlab code.\n\n\u003e\u003e syms  x y s0\n\u003e\u003e phi17=[3*x*(-x^2+3*x*y+2*x-1), 9*x*y*(x-y);\n                  9*x*y*(x-y), 3*y*(-3*x*y+4*x+y^2-1)];\n\u003e\u003e npn=[-sqrt(2)/2,-sqrt(2)/2]*phi17*[-sqrt(2)/2;-sqrt(2)/2];\n\u003e\u003e f=(1-s0)*subs(npn,{x,y},{1-s0,s0});\n\u003e\u003e sqrt(2)*integral(matlabFunction(f),0,1)\n\nans =\n\n      0.70711\n\n\u003e\u003e %which is equal to sqrt(2)/2, rather than 1.\n\nBesides, I used DefElement to verify the dual basis calculated by my code, this also suggests the missing sqrt(2).\n\n\n\n\n\n\n\n\n\n\n","author":{"url":"https://github.com/seinXiaoWei","@type":"Person","name":"seinXiaoWei"},"datePublished":"2025-12-11T05:09:19.000Z","interactionStatistic":{"@type":"InteractionCounter","interactionType":"https://schema.org/CommentAction","userInteractionCount":0},"url":"https://github.com/508/DefElement/issues/508"}