A finite element collocation method for quasilinear parabolic equations

Abstract:

Let the parabolic problem $c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 < x < 1,0 < t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)$, be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points ${\xi _{i,1}}$ and ${\xi _{i,2}}$ in subintervals $({x_{i - 1}},{x_i})$ for a function $U:[0,T] \to {\mathcal {H}_3}$, the class of Hermite piecewise-cubic polynomial functions with knots $0 = {x_0} < {x_1} < \cdots < {x_n} = 1$. It is shown that $u - U = O({h^4})$ uniformly in x and t, where $h = \max ({x_i} - {x_{i - 1}})$.