ABSTRACT. Although high-level software packages have greatly simplified the application of efficient finite element techniques to problems in flow (and beyond), far less attention has been applied to time stepping. Here, we describe the Irksome package, a high-level time stepping library for the Firedrake project. Users describe the semi-discrete variational form of their partial differential equation in an extension of the Unified Form Language (UFL). Irksome transforms this semi-discrete form into the fully discrete variational problem for a Runge-Kutta method at each time step. Irksome supports a wide range of RK methods, including explicit, diagonally implicit, and even fully implicit methods. Irksome inherits Firedrake's rich interface to PETSc, so we can also deploy novel preconditioners that make fully implicit methods competitive with and even superior to more widely-used schemes.

ABSTRACT. We discuss integration of the submesh feature in Firedrake that enables one to extract parts of an existing mesh and make them work together to solve finite element problems. We focus on cell submeshes, in which topological dimensions of the submeshes are the same as that of the parent mesh. We briefly discuss the implementation and the user interface, and then show interesting applications including fluid-structure interaction benchmark problem [Turek, S., Hron, J. (2006). Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow. In: Bungartz, HJ., Schäfer, M. (eds) Fluid-Structure Interaction. Lecture Notes in Computational Science and Engineering, vol 53. Springer, Berlin, Heidelberg].

ABSTRACT. Many classical and modern finite element spaces are derived by dividing each computational cell into finer pieces. Such macroelements frequently enable the enforcement of mathematically desirable properties such as divergence-free conditions or C1 continuity in a simpler or more efficient manner than elements without the subdivision. Although a few modern software projects provide one-off support for particular macroelements, a general approach facilitating broad-based support has, until now, been lacking. In this work, we describe a major addition to the FIAT project to support a wide range of different macroelements. These enhancements have been integrated into the Firedrake code stack. Numerical evaluation of the new macroelement facility is provided.

ABSTRACT. The solutions to partial differential equations frequently satisfy bounds constraints. When using finite element or finite difference methods, if one wishes to construct an approximate solution that satisfies these same bounds, great care is required. In a finite element context, one can replace a discrete variational problem with a discrete variational inequality. This allows for the selection of an approximate solution from a set of functions which satisfy the bounds constraints. Solving nonlinear optimization problems, though incurring a practical expense, bypasses known order barriers for linear problems and allows for the possibility of high-order and uniformly bounds-constrained finite element methods.

It is difficult to work with the entire set of bounds-constrained polynomials. However, the polynomials whose coefficients, when represented in the Bernstein basis, satisfy the bounds constraints form a convenient subset with which to work. Selecting an approximation from this set via a variational inequality, one obtains an approximation which is uniformly bounds-constrained, independent of the mesh used. Recent work seeks to extend this approach to collocation-type Runge-Kutta methods. Using a stage-value formulation, the collocating polynomial can be cast in the Bernstein basis to enforce bounds constraints uniformly in time. Examples are shown in which optimal order accuracy is observed.

ABSTRACT. We consider grad-div stabilized equal-order finite elements (FE) methods for the linearized Navier-Stokes equations. A block triangular preconditioner for the resulting system of algebraic equations is proposed which is closely related to the Augmented Lagrangian (AL) preconditioner. A field-of-values analysis of a preconditioned Krylov subspace method shows convergence bounds that are independent of the mesh parameter variation. Numerical studies support the theory and demonstrate the robustness of the approach also with respect to the viscosity parameter variation, as is typical for AL preconditioners when applied to inf-sup stable FE pairs. The numerical experiments also address the accuracy of grad-div stabilized equal-order FE method for the steady state Navier-Stokes equations.

ABSTRACT. We consider the interaction between a (poro)elastic structure and a free-flowing fluid. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the fluid and structure regions. The phase field function continuously transitions from one to zero in a diffuse region of width order epsilon around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretisation of the diffuse interface model to the continuous diffuse model. We in turn prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis. We also discuss recent work extending this to the large deformation, moving domain case.