ABSTRACT. We present a doubly enriched finite volume method for precisely computing highly dynamic fluid-particle interaction. This involves forces beeing exchanged between the particles and the fluid at the interface. In an earlier work we derived a monolithic scheme which includes the interaction forces and rigied-body motions into the Navier-Stokes equations by extending the test space. In highly dynamic particle-fluid interaction cases, pressure oscillations are a common issue. The surface force term yields a jump in the pressure and therefore the oscillations are usually resolved by extending the spaces on cut elements in order to resolve the discontinuity. For the construction of the enriched spaces proposed in this talk we exploit the Petrov-Galerkin formulation of the vertex-centered finite volume method (PG-FVM). From the perspective of the finite volume scheme we argue that wrong discrete normal directions at the interface are the origin of the oscillations. To cope with these oscillations, we additionally extended the ansatz space by what we call flat-top functions, which are cut off at the interface. We tested our doubly extended method with a falling ellipsis problem and the well-known Drafting-Kissing-Tumbling (DKT) problem by Joseph. The results are convincing wrt. speed and precision. Computations were carried out in parallel using the UG4 simulation system.

ABSTRACT. In this work, we build upon multilevel methods initially developed for linear-quadratic distributed optimal control of elliptic equations, applying them to the classical problem of identifying the diffusivity in a heat transfer problem from various state measurements. Specifically, given synthetic experiments involving the application of heat sources and multiple measurements of temperature and heat fluxes, the objective is to determine the diffusivity tensor, which, for the purposes of this talk, is assumed to be isotropic. Due to the ill-posed nature of this problem, it is formulated as a regularized least-squares problem, which is both nonlinear and non-convex. We employ a hierarchy of geometric grids and multilevel techniques to precondition the Hessian used in the Newton-CG solution process, in the absence of constraints, or the semi-smooth Newton-CG method when bound constraints are applied to the diffusivity.

ABSTRACT. MGARD (MultiGrid Adaptive Reduction of Data) is a compression and data refactoring algorithm grounded in multigrid method principles. Its foundation lies in stable multilevel decompositions of conforming piecewise linear finite element spaces, facilitating precise error control across multiple norms and derived quantities of interest. This work extends the approach to encompass Lagrange finite elements of arbitrary order and reinterprets the algorithm as a lifting scheme incorporating polynomial predictors of varying orders. Furthermore, a novel formulation employing a compactly supported wavelet basis is introduced, along with an explicit construction of the corresponding wavelet transform for uniform dyadic grids.

ABSTRACT. Despite the fact that Parallel-in-Time (PinT) methods are predicted to become necessary to fully utilize next-generation exascale machines, there are currently no known practical methods which scale well with the length of the time-domain for chaotic problems, due to exponential dependence of the condition number on the fastest chaotic time-scale. While most prior works applying multigrid-in-time to chaotic systems focus on the coarse grid equation, we investigate the relaxation techniques commonly used and demonstrate that they in fact diverge for chaotic systems. Here the novel Local Shadowing Relaxation (LSR) is presented and proven to be a convergent, PinT relaxation for chaotic PDE systems. Promising preliminary analytical results and numerical experiments with the Lorenz system indicate that LSR may solve the scaling problem for chaotic systems, potentially allowing space-time parallelization of turbulent computational fluid dynamics.

ABSTRACT. Multi-Grid Reduction in Time (MGRIT) is a well-established parallel-in-time integration technique that employs a multi-grid framework to accelerate the time-to-solution for a system of ordinary differential equations. Our recent work modifies MGRIT by using the deferred correction framework to improve accuracy at almost no additional computational cost. Unlike classical MGRIT, which often solves a rediscretized coarse grid problem on coarser meshes, MGRIT-DC instead solves the deferred correction error equation, leveraging information readily available from finer meshes. In this presentation, we introduce the MGRIT-DC method and give some preliminary analysis. Numerical experiments demonstrate its effectiveness in enhancing the solution accuracy across various test problems.

ABSTRACT. We formulate and investigate a parallel-in-time method based on least-squares and multigrid principles. Derived from a normal-equations formulation of a semi-discrete partial differential equation (PDE), this approach shows potential in accelerating the solution of PDEs. However, it currently requires high processor counts and problems with stringent accuracy demands. We explore several methods of speeding up the approach and identify promising avenues for future exploration.