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 x x + b ( x , t , u , u x ) , 0 > x > 1 , 0 > t ≦ T , u ( x , 0 ) = f ( x ) , u ( 0 , t ) = g 0 ( t ) , u ( 1 , t ) = g 1 ( t ) 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 ξ i , 1 {\xi _{i,1}} and ξ i , 2 {\xi _{i,2}} in subintervals ( x i − 1 , x i ) ({x_{i - 1}},{x_i}) for a function U : [ 0 , T ] → H 3 U:[0,T] \to {\mathcal {H}_3} , the class of Hermite piecewise-cubic polynomial functions with knots 0 = x 0 > x 1 > ⋯ > x n = 1 0 = {x_0} > {x_1} > \cdots > {x_n} = 1 . It is shown that u − U = O ( h 4 ) u - U = O({h^4}) uniformly in x and t, where h = max ( x i − x i − 1 ) h = \max ({x_i} - {x_{i - 1}}) .