Abstract: High order finite element methods can solve partial differential equations more efficiently than low order methods. But how large of a polynomial degree is beneficial? This paper addresses that question through a case study of three problems representing problems with smooth solutions, problems with steep gradients, and problems with singularities. It also contrasts h-adaptive, p-adaptive, and h-adaptive refinement. The results indicate that for low accuracy requirements, like 1% relative error, h-adaptive refinement with relatively low order elements is sufficient, and for high accuracy requirements, p-adaptive refinement is best for smooth problems and hp-adaptive refinement with elements up to about 10th degree is best for other problems.

Abstract: Parametric ordinary differential equations (ODE) arise in many engineering applications. We consider ODE solutions to be embedded in an overall objective function which is to be minimized, e.g. for parameter estimation. For derivative-based optimization algorithms adjoint methods should be used. In this article, we present a discrete adjoint ODE integration framework written in C++ (NIXE 2.0) combined with Algorithmic Differentiation by overloading (dco/c++). All required derivatives, i.e. Jacobians for the integration as well as gradients and Hessians for the optimization, are generated automatically. With this framework, derivatives of arbitrary order can be implemented with minimal programming effort. The practicability of this approach is demonstrated in a dynamic parameter estimation case study for a batch fermentation process using sequential method of dynamic optimization. Ipopt is used as the optimizer which requires second derivatives.

Abstract: Krylov methods preconditioned by Factorized Sparse Approximate Inverses (FSAI) are an efficient approach for the solution of symmetric positive definite linear systems on massively parallel computers. However, FSAI often suffers from a high set-up cost, especially in ill-conditioned problems. In this communication we propose a novel algorithm for the FSAI computation that makes use of the concept of supernode borrowed from sparse LU factorizations and direct methods.

Abstract: We analyze a possibility of turning off post-smoothing(relaxation) in geometric multigrid when used as a preconditioner in preconditioned conjugate gradient (PCG) linear and eigenvalue solvers for the 3D Laplacian. The geometric Semicoarsening Multigrid (SMG) method is provided by the hypre parallel software package. We solve linear systems using two variants (standard and flexible) of PCG and preconditioned steepest descent (PSD) methods. The eigenvalue problems are solved using the locally optimal block preconditioned conjugate gradient (LOBPCG) method available in hypre through BLOPEX software. We observe that turning off the post-smoothing in SMG dramatically slows down the standard PCG-SMG. For flexible PCG and LOBPCG, our numerical tests show that removing the post-smoothing results in overall 40--50 percent acceleration, due to the high costs of smoothing and relatively insignificant decrease in convergence speed. We demonstrate that PSD-SMG and flexible PCG-SMG converge similarly if SMG post-smoothing is off. A theoretical justification is provided.