A Finite Element Collocation Method for Quasilinear Parabolic Equations

Let the parabolic problem <!-- MATH $c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 < x < 1,0 < t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)$ --> <img width="939" height="41" align="MIDDLE" border="0" src="images/img1.gif" alt="$ 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 <!-- MATH ${\xi _{i,1}}$ --> and <!-- MATH ${\xi _{i,2}}$ --> in subintervals <!-- MATH $({x_{i - 1}},{x_i})$ --> for a function <!-- MATH $U:[0,T] \to {\mathcal{H}_3}$ --> , the class of Hermite piecewise-cubic polynomial functions with knots <!-- MATH $0 = {x_0} < {x_1} < \cdots < {x_n} = 1$ --> <img width="245" height="37" align="MIDDLE" border="0" src="images/img6.gif" alt="$ 0 = {x_0} < {x_1} < \cdots < {x_n} = 1$">. It is shown that <!-- MATH $u - U = O({h^4})$ --> uniformly in x and t, where <!-- MATH $h = \max ({x_i} - {x_{i - 1}})$ --> .