Abstract. We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs-Boltzmann distribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with O(N ) computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications. Introduction. We develop general computational methods for applications involving the microscopic mechanics of spatially extended elastic bodies within a fluid that are subjected to thermal fluctuations. Motivating applications include the study of the microstructures of complex fluids [17], lipid bilayer membranes [28,32,48], and micro-mechanical devices [37,29]. Even in the deterministic setting, the mechanics of fluid-structure interactions pose a number of difficult and long-standing challenges owing to the rich behaviors that can arise from the interplay of the fluid flow and elastic stresses of the microstructures [19,42]. To obtain descriptions tractable for analysis and simulations, approximations are often introduced into the fluid-structure coupling. For deterministic systems, many spatially adaptive numerical methods have been developed for approximate fluid-structure interactions [25,35,26,39,2,30]. In the presence of thermal fluctuations, additional challenges arise from the need to capture in computational methods the appropriate propagation of fluctuations throughout the discretized system to obtain results consistent with statistical mechanics. In practice, challenges arise from the very different dissipative properties of the discrete operators relative to their continuum differential counterparts. These issues have important implications for how stochastic fluctuations should be handled in the discrete setting. Even when it is possible to formulate stochastic driving fields in a well-founded manner consistent with statistical mechanics, these Gaussian random

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Abstract. With the rise in popularity of compatible finite element, finite difference and finite volume discretizations for the time domain eddy current equations, there has been a corresponding need for fast solvers of the resulting linear algebraic systems. However, the traits that make compatible discretizations a preferred choice for the Maxwell's equations also render these linear systems essentially intractable by truly black-box techniques. We propose a new algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid technique (AMG) for this reformulated problem. The reformulation process takes advantage of a discrete Hodge decomposition on co-chains to replace the discrete eddy current equations by an equivalent 2 × 2 block linear system whose diagonal blocks are discrete Hodge Laplace operators acting on 1-cochains and 0-cochains, respectively. While this new AMG technique requires somewhat specialized treatment on the finest mesh, the coarser meshes can be handled using standard methods for Laplace-type problems. Our new AMG method is applicable to a wide range of compatible methods on structured and unstructured grids, including edge finite elements, mimetic finite differences, co-volume methods and Yee-like schemes. We illustrate the new technique, using edge elements in the context of smoothed aggregation AMG, and present computational results for problems in both two and three dimensions.Key words. Maxwell's equations, eddy currents, algebraic multigrid, multigrid, discrete Hodge Laplacian, compatible discretizations, edge elements. AMS subject classifications. 65F10, 65F30, 78A301. Introduction. Due to the pioneering work of Bossavit [9], it is now wellknown that stable and accurate numerical solution of Maxwell's equations can be achieved by using discrete spaces from a finite-dimensional analogue of the differential De Rham complex. Numerical methods that are based on such discretizations are now commonly referred to as compatible, or mimetic methods [6].However, the same traits that make compatible methods a natural choice for Maxwell's equations also render the ensuing linear system essentially intractable by truly general purpose black-box multigrid solvers. For example, a compatible discretization of the curl-curl operator gives rise to a symmetric, semi-definite linear system whose null-space is of approximately the same size as the number of nodes in the computational grid. Multilevel solution of such systems often utilizes special smoothing, prolongation and restriction operators that separate error components in the null-space and its complement and satisfy a commuting diagram property. Formulation of such operators is well-understood in geometric multigrid settings [22], where

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A template-based generic programming approach was presented in a previous paper [19] that separates the development effort of programming a physical model from that of computing additional quantities, such as derivatives, needed for embedded analysis algorithms. In this paper, we describe the implementation details for using the template-based generic programming approach for simulation and analysis of partial differential equations (PDEs). We detail several of the hurdles that we have encountered, and some of the software infrastructure developed to overcome them. We end with a demonstration where we present shape optimization and uncertainty quantification results for a 3D PDE application.

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We consider the sequence of sparse matrix-matrix multiplications performed during the setup phase of algebraic multigrid. In particular, we show that the most commonly used parallel algorithm is often not the most communication-efficient one for all of the matrix-matrix multiplications involved. By using an alternative algorithm, we show that the communication costs are reduced (in theory and practice), and we demonstrate the performance benefit for both model (structured) and more realistic unstructured problems on large-scale distributed-memory parallel systems. Our theoretical analysis shows that we can reduce communication by a factor of up to 5.4 for a model problem, and we observe in our empirical evaluation communication reductions of factors up to 4.7 for structured problems and 3.7 for unstructured problems. These reductions in communication translate to run-time speedups of up to factors of 2.3 and 2.5, respectively.