I want to discretize (PDE) non-linear heat equation using Galerkin FEM. it includes non-linear reaction reaction rate which can be solved by numerical technique only. please guide me how to approximate 'T' using 1-D linear hat function?
I recommend reading the help of the ANSYS CFX which has a wonderful description of all methods with equations :
I recommend reading the help of the ANSYS CFX which has a wonderful description of all methods with equations :
Please go through the following papers (thesis/journals):
http://www.math.chalmers.se/~mohammad/teaching/PDEbok/Draft_I+II.pdf
http://epubs.siam.org/doi/abs/10.1137/0707048
http://www.mathunion.org/ICM/ICM1978.2/Main/icm1978.2.0943.0952.ocr.pdf
I invite you to read the content in this link
http://www.mssmat.ecp.fr/files/content/sites/mssmat/files/Personnel/clouteaud/lecture3.pdf
and try to download the attached book (800 pages) (6.95 MB)
i couldn't find \delta_ij in your equation " R".
you should rewrite your equation as a summation on i=0..n and then solve each of integrals in interval " x_ i to x_ i+1" by any of numerical methods that you want.
I think there's a couple of typos in the integrand of R1. First of all, R1 must be a vector of which the i-th component is given by your equation. Secondly in the denominator of the exponent, T(t) should have a subscript i, otherwise you could take it out of the summation sign.
Now you must realize, that \phi_i is only nonzero on two elements, e_(i-1) and e_i. The only non zero basis functions on those elements are \phi_(i-1) and \phi_i on element e_(i-1) and \phi_i and \phi_(i+1) on e_i. So your integral simplifies to
R1_i(t)=\int_e_(i-1) exp (c/(T_i-1(t)\phi_(i-1)+T_i\phi_i)) phi_i dx +
\int_e_I exp (c/(T_i\phi_i + T_(i+1) \phi_(i+1))) \phi_i dx
c=-6039
Simplest would be to apply the trapezoidal rule giving
R1_i(t) = 0.5 (dx_(i-1) + dx_i) exp (c/T_i(t))
alternativly you could linearly interpolate the exponent on each element, i.e.
exp (c/(T_(i-1)\phi_(i-1) + T_i\phi_i)) \approx exp(c/T_(i-1))\phi_(i-1) + exp(c/T_i)\phi_i
on e_i-1 and a similar expression on e_i and take it from there, since you can calculate \int_e_(i-1) \phi_(i-1)\phi_i dx etc exactly.
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