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Bayesian methods have been widely used in the last two decades to infer statistical proper- ties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases the number of variables used to param- eterize these coefficients is large, and oobtaining meaningful statistics of their probability distributions is difficult using simple sampling methods such as the basic Metropolis– Hastings algorithm—in particular, if the inverse problem is ill-conditioned or ill-posed. As a consequence, many advanced sampling methods have been described in the literature that converge faster than Metropolis–Hastings, for example, by exploiting hierarchies of statistical models or hierarchies of discretizations of the underlying differential equation. At the same time, it remains difficult for the reader of the literature to quantify the advantages of these algorithms because there is no commonly used benchmark. This paper presents a benchmark Bayesian inverse problem—namely, the determination of a spatially variable coefficient, discretized by 64 values, in a Poisson equation, based on point mea- surements of the solution—that fills the gap between widely used simple test cases (such as superpositions of Gaussians) and real applications that are difficult to replicate for de- velopers of sampling algorithms. We provide a complete description of the test case and provide an open-source implementation that can serve as the basis for further experiments. We have also computed 2 × 10^11 samples, at a cost of some 30 CPU years, of the poste- rior probability distribution from which we have generated detailed and accurate statistics against which other sampling algorithms can be tested. 

Thehp-adaptive finite element method—where one independently chooses the mesh size (h) and polynomial degree (p) to be used on each cell—has long been known to have better theoretical convergence properties than eitherh- orp-adaptive methods alone. However, it is not widely used, owing at least in part to the difficulty of the underlying algorithms and the lack of widely usable implementations. This is particularly true when used with continuous finite elements.

Herein, we discuss algorithms that are necessary for a comprehensive and generic implementation ofhp-adaptive finite element methods on distributed-memory, parallel machines. In particular, we will present a multistage algorithm for the unique enumeration of degrees of freedom suitable for continuous finite element spaces, describe considerations for weighted load balancing, and discuss the transfer of variable size data between processes. We illustrate the performance of our algorithms with numerical examples and demonstrate that they scale reasonably up to at least 16,384 message passage interface processes.

We provide a reference implementation of our algorithms as part of the open source librarydeal.II.

 

Abstract. Geodynamical simulations over the past decades have widely beenbuilt on quadrilateral and hexahedral finite elements. For thediscretization of the key Stokes equation describing slow, viscousflow, most codes use either the unstable Q1×P0 element, astabilized version of the equal-order Q1×Q1 element, ormore recently the stable Taylor–Hood element with continuous(Q2×Q1) or discontinuous (Q2×P-1)pressure. However, it is not clear which of these choices isactually the best at accurately simulating “typical” geodynamicsituations. Herein, we provide a systematic comparison of all of theseelements for the first time. We use a series of benchmarks that illuminate differentaspects of the features we consider typical of mantle convectionand geodynamical simulations. We will show in particular that the stabilizedQ1×Q1 element has great difficulty producing accuratesolutions for buoyancy-driven flows – the dominant forcing formantle convection flow – and that the Q1×P0 element istoo unstable and inaccurate in practice. As a consequence, webelieve that the Q2×Q1 and Q2×P-1 elementsprovide the most robust and reliable choice for geodynamical simulations,despite the greater complexity in their implementation and thesubstantially higher computational cost when solving linearsystems. 

The traditional workflow in continuum mechanics simulations is that a geometry description —for example obtained using Constructive Solid Geometry (CSG) or Computer Aided Design (CAD) tools—forms the input for a mesh generator. The mesh is then used as the sole input for the finite element, finite volume, and finite difference solver, which at this point no longer has access to the original, “underlying” geometry. However, many modern techniques—for example, adaptive mesh refinement and the use of higher order geometry approximation methods—really do need information about the underlying geometry to realize their full potential. We have undertaken an exhaustive study of where typical finite element codes use geometry information, with the goal of determining what information geometry tools would have to provide. Our study shows that nearly all geometry-related needs inside the simulators can be satisfied by just two “primitives”: elementary queries posed by the simulation software to the geometry description. We then show that it is possible to provide these primitives in all of the frequently used ways in which geometries are described in common industrial workflows, and illustrate our solutions using a number of examples. 

Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.4. 

Abstract This paper provides an overview of the new features of the finite element library deal.II , version 9.5. 

Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.3.