A variationally separable splitting for the generalized-$α$ method for parabolic equations
This work provides an efficient solver for parabolic PDEs discretized with finite elements or isogeometric analysis, but it is an incremental improvement over existing splitting methods.
The authors developed a variationally separable splitting technique for the generalized-α method that achieves linear computational cost in degrees of freedom for multi-dimensional parabolic equations, while maintaining second-order accuracy in time and optimal spatial convergence rates.
We present a variationally separable splitting technique for the generalized-$α$ method for solving parabolic partial differential equations. We develop a technique for a tensor-product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multi-dimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user-controlled high-frequency dissipation while minimizing unwanted low-frequency dissipation. The method has second-order accuracy in time and optimal rates ($h^{p+1}$ in $L^2$ norm and $h^p$ in $L^2$ norm of $\nabla u$) in space. We present the spectrum analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.