KEYWORDScubic spline, H 1 -Galerkin mixed method, Kuramoto-Sivashinsky equation, optimal-order error estimates, semi-discrete and fully discrete schemes, instability of solution Classification Codes. 65N30; 65N15; 65M60; 65M15; 35G20.Numer Methods Partial Differential Eq. 2019;35:445-477. wileyonlinelibrary.com/journal/num