The Complementary Subspaces Method and Its Application to Numerical Solution of the Equations of Convection

The complementary subspaces (CS) method is aimed at numerically solving equations of the form $P_{\cal U}^{}(Av-f)=0$ emerging when evolutionary equations are integrated by spectral methods. Here $v$ must belong to the finite-dimensional space ${\cal V}$ comprised of functions satisfying the prescribed boundary conditions and $P_{\cal U}^{}$ is an orthogonal projection on a finite-dimensional space ${\cal U}$. The idea of the CS method is to find the solution to the original problem by solving a~modified problem, $P_{\cal G}^{}(Aw-\widetilde f)=0$, $w\in{\cal W}$, and either adding the correction $\widetilde f-f$ before solving the modified problem, or computing the correction $v-w$ afterwards. The spaces ${\cal W}$ and ${\cal G}$ are chosen in such a way that solving the modified problem requires less operations than the original one. The method was introduced in [23]; we propose now its modification allowing more freedom in choosing the spaces for the modified problem and in computing the correction. The algorithm is discussed in the general form and in the case of the spaces spanned by linear combinations of Chebyshev polynomials. The method is applied for numerical investigation of convective flows in a plane horizontal layer heated from below and rotating about an inclined axis with no-slip horizontal boundaries. For the employed values of control parameters the temporal behavior is comprised of repeating events and is possibly related to heteroclinic connections between unstable steady states.