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International audience; The aim of this paper is to propose efficient weak coupling formulations between the boundary element method and the high-order finite element method for solving time-harmonic electromagnetic scattering problems. The approach is based on the use of a non-overlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Padé approximants of the nonlocal Magnetic-to-Electric operators. Numerical results are presented to validate and analyze the new approach for both homogeneous and inhomogeneous scatterers.

We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis-functions yields cheaply invertible block-diagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence., 34 pages, 14 figures, submitted

A convergence study of the forward problem of electrical impedance tomography is performed using triangular high-order piecewise polynomial finite-element methods (p-FEM) on a square domain. The computation of p-FEM for the complete electrode model (CEM) is outlined and a novel analytic solution to the CEM on a square domain is presented. Errors as a function of mesh-refinement and computational time, as well as convergence rates as a function of contact impedance, are computed numerically for different polynomial approximation orders. It is demonstrated that p-FEM can generate more accurate forward solutions in less computational time, which implies more accurate simulated interior potentials, electrode voltages and conductivity Jacobians.

International audience; The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a non-iterative method for computing intersections by solving a matrix singular value decomposition (SVD) and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one-shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton-Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree $q$ of the shape functions and is $2q \times 3q$ for curves and $6 q^2 \times 8 q^2$ for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used high-order finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitro-project/.

We present a numerical discretisation of an embedded two-dimensional manifold using high-order continuous Galerkin spectral/hp elements, which provide exponential convergence of the solution with increasing polynomial order, while retaining geometric flexibility in the representation of the domain. Our work is motivated by applications in cardiac electrophysiology where sharp gradients in the solution benefit from the high-order discretisation, while the computational cost of anatomically-realistic models can be significantly reduced through the surface representation and use of high-order methods. We describe and validate our discretisation and provide a demonstration of its application to modelling electrochemical propagation across a human left atrium.

As computing hardware evolves, increasing core counts mean that memory bandwidth is becoming the deciding factor in attaining peak performance of numerical methods. High-order finite element methods, such as those implemented in the spectral/hp framework Nektar++, are particularly well-suited to this environment. Unlike low-order methods that typically utilise sparse storage, matrices representing high-order operators have greater density and richer structure. In this paper, we show how these qualities can be exploited to increase runtime performance on nodes that comprise a typical high-performance computing system, by amalgamating the action of key operators on multiple elements into a single, memory-efficient block. We investigate different strategies for achieving optimal performance across a range of polynomial orders and element types. As these strategies all depend on external factors such as BLAS implementation and the geometry of interest, we present a technique for automatically selecting the most efficient strategy at runtime.

In this paper, we present a comprehensive framework for the simulation of Multifluid flows based on the implicit level-set representation of interfaces and on an efficient solving strategy of the Navier-Stokes equations. The mathematical framework relies on a modular coupling approach between the level-set advection and the fluid equations. The space discretization is performed with possibly high-order stable finite elements while the time discretization features implicit Backward Differentation Formulae of arbitrary order. This framework has been implemented within the Feel++ library, and features seamless distributed parallelism with fast assembly procedures for the algebraic systems and efficient preconditioning strategies for their resolution. We also present simulation results for a three-dimensional Multifluid benchmark, and highlight the importance of using high-order finite elements for the level-set discretization for problems involving the geometry of the interfaces, such as the curvature or its derivatives.

This paper presents high-order implementations of a generalized finite element method for through-the-thickness three-dimensional branched cracks. This approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks, independently of the background finite element mesh. Representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies. The combination of local refinement at crack fronts and high-order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions. Copyright © 2007 John Wiley & Sons, Ltd.

International audience; A general method for the post-processing treatment of high-order finite element fields is presented. The method applies to general polynomial fields, including discontinuous finite element fields. The technique uses error estimation and h-refinement to provide an optimal visualization grid. Some filtering is added to the algorithm in order to focus the refinement on a visualization plane or on the computation of one single iso-zero surface. 2D and 3D examples are provided that illustrate the power of the technique. In addition, schemes and algorithms that are discussed in the paper are readily available as part of an open source project that is developed by the authors, namely Gmsh.