A Phase-Space Discontinuous Galerkin Scheme for the Radiative Transfer Equation in Slab Geometry

We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging

Convergence and rate optimality of adaptive multilevel stochastic Galerkin FEM

We analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in two-dimensional physical domains, with coefficients and right-hand-side functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos expansion with coefficients residing in finite element spaces associated with different locally refined meshes. Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximation spaces

Adaptive algorithms for partial differential equations with parametric uncertainty

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