Adaptive algorithms for partial differential equations with parametric uncertainty

Rocchi, Leonardo (2019). Adaptive algorithms for partial differential equations with parametric uncertainty. University of Birmingham. Ph.D.

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In this thesis, we focus on the design of efficient adaptive algorithms for the numerical approximation of solutions to elliptic partial differential equations (PDEs) with parametric inputs. Numerical discretisations are obtained using the stochastic Galerkin Finite Element Method (SGFEM) which generates approximations of the solution in tensor product spaces of finite element spaces and finite-dimensional spaces of multivariate polynomials in the random parameters.

Firstly, we propose an adaptive SGFEM algorithm which employs reliable and efficient hierarchical a posteriori energy error estimates of the solution to parametric PDEs. The main novelty of the algorithm is that a balance between spatial and parametric approximations is ensured by choosing the enhancement associated with dominant error reduction estimates.

Next, we introduce a two-level a posteriori estimate of the energy error in SGFEM approximations. We prove that this error estimate is reliable and efficient. Then we provide a rigorous convergence analysis of the adaptive algorithm driven by two-level error estimates. Four different marking strategies for refinement of stochastic Galerkin approximations are proposed and, in particular, for two of them, we prove that the sequence of energy errors computed by associated algorithms converges linearly.

Finally, we use duality techniques for the goal-oriented error estimation in approximating linear quantities of interest derived from solutions to parametric PDEs. Adaptive enhancements in the proposed algorithm are guided by an innovative strategy that combines the error reduction estimates computed for spatial and parametric components of corresponding primal and dual solutions.

The performance of all adaptive algorithms and the effectiveness of the error estimation strategies are illustrated by numerical experiments. The software used for all experiments in this work is available online.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Licence: All rights reserved
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: Engineering and Physical Sciences Research Council
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Q Science > QA Mathematics > QA76 Computer software
Q Science > QC Physics


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