As a key step towards a complete automation of the finite element method, we present a new algorithm for automatic and efficient evaluation of multilinear variational forms. The algorithm has been implemented in the form of a compiler, the FEniCS Form Compiler FFC. We present benchmark results for a series of standard variational forms, including the incompressible Navier– Stokes equations and linear elasticity. The speedup compared to the standard quadrature-based approach is impressive; in some cases the speedup is as large as a factor 1000.

Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart–Thomas finite element method with a reduction factor ρ<1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity. 1.

We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry and variable coefficients. Based on this representation theorem, we design an algorithm for efficient pretabulation of the reference tensor. The new algorithm has been implemented in the FEniCS Form Compiler (FFC) and improves on a previous loop-based implementation by several orders of magnitude, thus shortening compile-times and development cycles for users of FFC. Categories and Subject Descriptors: G.4 [Mathematical Software]—Algorithm design and analysis,

Abstract. We consider the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of two materials coupled through a common boundary. We derive accurate a posteriori error estimates that account for the transfer of error between components of the operator decomposition method as well as the differences between the adjoints of the full problem and the discrete iterative system. We use these estimates to guide adaptive mesh refinement. In addition, we address a loss of order of convergence that results from the decomposition, and show that the approximation order of convergence is limited by the accuracy of the transferred gradient information. We employ a boundary flux recovery method to regain the expected order of accuracy in an efficient manner. Key words. a posteriori error analysis, adaptive mesh refinement, adjoint problem, boundary flux method, conjugate heat transfer, domain decomposition, finite element method, generalized

Abstract not found

Abstract. We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method. 1.

We propose a new method for limiting the gradients in a mesh size function by solving a non-linear partial differential equation on the background mesh. Our gradient limiting Hamilton-Jacobi equation simplifies the generation of mesh size functions significantly, by decoupling size constraints at specific locations from the mesh grading requirements. We derive an analytical solution for convex domains which shows the results are optimal, and we describe how to implement efficient solvers on various types of meshes. We demonstrate our size functions with a proposed new mesh generation algorithm, using examples with curvature, feature size, and numerical adaptation.

Abstract not found

Mesh generation and mesh enhancement algorithms often require a mesh size function to specify the desired size of the elements. We present algorithms for automatic generation of a size function, discretized on a background grid, by using distance functions and numerical PDE solvers. The size function is adapted to the geometry, taking into account the local feature size and the boundary curvature. It also obeys a grading constraint that limits the size ratio of neighboring elements. We formulate the feature size in terms of the medial axis transform, and show how to compute it accurately from a distance function. We propose a new Gradient Limiting Equation for the mesh grading requirement, and we show how to solve it numerically with Hamilton-Jacobi solvers. We show examples of the techniques using Cartesian and unstructured background grids in 2-D and 3-D, and applications with numerical adaptation and mesh generation for images. Keywords: mesh generation, size function, background grid, Hamilton-Jacobi, gradation control 1.

Two parallel multilevel methods are presented for solving large discretized partial differential equations on unstructured 2D/3D grids. In short, the presented methods combine three powerful numerical algorithms, i.e., overlapping domain decomposition, multigrid method and adaptivity. As the foundation of the methods we propose an algorithm for generating and partitioning a hierarchy of adaptively refined unstructured grids, so that adaptivity can be incorporated up to a certain grid level and the resulting subgrids are well-balanced. The first method uses a domain decomposition approach where we replace the single-level coarse grid in standard two level overlapping Schwarz methods by a hierarchy of adaptively refined unstructured grids. Efficient coarse grid correction is done by multigrid V-cycles. Moreover, multigrid V-cycles are also used in local subproblem solves. The second method uses a multigrid approach where we parallelize global multigrid V-cycles on all the grid levels. Nu...