A finite element collocation method for quasilinear parabolic equations
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- by Jim Douglas and Todd Dupont PDF
- Math. Comp. 27 (1973), 17-28 Request permission
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 $h = \max ({x_i} - {x_{i - 1}})$.References
- Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1963. MR 0157156
- Jim Douglas Jr. and Todd Dupont, The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures, Math. Comput. 20 (1975), no. 130, 360–389. MR 0502033, DOI 10.1090/S0025-5718-1975-0502033-7
- J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243 B. Wendroff, First Principles of Numerical Analysis, Addison-Wesley, Reading, Mass., 1969. M. F. Wheeler, Thesis, Rice University, Houston, Texas, 1971; SIAM J. Numer. Anal. (To appear.)
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp. 27 (1973), 17-28
- MSC: Primary 65N05
- DOI: https://doi.org/10.1090/S0025-5718-1973-0339508-8
- MathSciNet review: 0339508