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  • $\begingroup$ A possible work-around just come to my mind: axx = Inverse[mat\[Phi]\[Phi]] . (mat\[Phi]\[Phi]x) . Inverse[mat\[Phi]\[Phi]] . (mat\[Phi]\[Phi]x) . f1i :) $\endgroup$ Commented Dec 16, 2024 at 10:06
  • $\begingroup$ @xzczd Thanks for your clever workaround. Works fine! How do you justify this workaround $\endgroup$ Commented Dec 16, 2024 at 10:18
  • $\begingroup$ I'm first inspired by this observation of mine. Then, by checking Partial Differential Equations and Boundary Conditions section of the tutorial Solving Partial Differential Equations with Finite Elements, I notice the discretization of convection term doesn't seem to involve integration by parts i.e. the zero Neumann value isn't imposed when a pure first order derivative (e.g. $\frac\partial{\partial x}$) is discretized. $\endgroup$ Commented Dec 16, 2024 at 10:32
  • $\begingroup$ @xzczd Very clever. I'm just trying the 2D workaround, actually without succes $\endgroup$ Commented Dec 16, 2024 at 10:35
  • $\begingroup$ $\frac{\partial^2}{\partial x \partial y}$ can be calculated like this: {mat\[Phi]\[Phi]x, mat\[Phi]\[Phi]y} = (initCoeffs = InitializePDECoefficients[vd, sd, "ConvectionCoefficients" -> {{{#}}}]; methodData = InitializePDEMethodData[vd, sd]; DiscretizePDE[initCoeffs, methodData, sd]["StiffnessMatrix"]) & /@ {{1, 0}, {0, 1}};f2 = Function[{x, y}, Sin[2 Pi (x - Pi/2)] Cos[Pi y]]; func = LinearSolve[mat\[Phi]\[Phi]]; axy = func[mat\[Phi]\[Phi]y . func[mat\[Phi]\[Phi]x . Map[Apply[f2, #] &, punkte]]]; but still, the precision isn't good enough… (The simple $f(x,y)=x y$ case is good, though. ) $\endgroup$ Commented Dec 17, 2024 at 11:53