“If I’ll not use Method->”EquationSimplification”-> “Residual”, will have some addtional error as well.”
Needs["VectorAnalysis`"] f = 1/2 K1 Dot[s, s] + 1/2 K2 (t)^2 + 1/2 K3 Dot[b, b] - (ξo ξv (E1)^2 )/ 2 - (ξo ξa (E1)^2 (n)^2)/2 n = {Cos[θ[z]], 0, Sin[θ[z]]}; s = n Div[n, Cartesian[x, y, z]]; t = Dot[n, Curl[n, Cartesian[x, y, z]]]; b = Cross[n, Curl[n, Cartesian[x, y, z]]]; ele=\!\( \*SubscriptBox[\(\[PartialD]\), \(\[Theta][z]\)]f\)-\!\( \*SubscriptBox[\(\[PartialD]\), \(z\)]\((\* SubscriptBox["\[PartialD]", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "z", "]"}]]f)\)\) 
K1 = 1; K3 = 0.5 K1; ξo = 1; ξa = 1; L = 1; E1 = 1; ele nd = NDSolve[{ele == 0, θ[0] == 0, θ[L] == Pi/2}, θ[z], {z, 0, 1}, Method -> {"EquationSimplification" -> "Residual"}] 
