Least-squares proper generalised decompositions for weakly coercive elliptic problems
Croft, Thomas L. D.
Phillips, Timothy Nigel
- Publisher: Society for Industrial and Applied Mathematics
arxiv: Mathematics::Numerical Analysis
Proper generalised decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric and strongly coercive. In the\ud particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem\ud as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two different reformulations of the problem. We show that these least-squares PGD provide a much stabler algorithm than an equivalent Galerkin PGD and provide proofs of convergence of the algorithms.