Abstract
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.
| Original language | English |
|---|---|
| Pages (from-to) | 2359-2382 |
| Number of pages | 24 |
| Journal | SIAM Journal on Numerical Analysis |
| Volume | 57 |
| Issue number | 5 |
| Early online date | 3 Oct 2019 |
| DOIs | |
| Publication status | Published - 3 Oct 2019 |
Funding
\ast Received by the editors December 4, 2018; accepted for publication (in revised form) July 25, 2019; published electronically October 3, 2019. https://doi.org/10.1137/18M1229560 Funding: The work of the first author was supported by the EPSRC under grant EP/P013791/1 and by The Alan Turing Institute under the EPSRC grant EP/N510129/1. The work of the second and fourth authors was supported by the Austrian Science Fund (FWF) under grants W1245 and F65. \dagger School of Mathematics, University of Birmingham, Birmingham, UK ([email protected], [email protected]). \ddagger Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria (dirk.praetorius@ asc.tuwien.ac.at). \S Faculty of Mathematics, University of Vienna, Vienna, Austria ([email protected]).
Keywords
- a posteriori error analysis
- adaptive methods
- finite element methods
- parametric PDEs
- stochastic Galerkin methods
- two-level error estimate