A Numerical Approach to Space-Time Finite Elements for the Wave Equation

We study a space-time finite element approach for the nonhomogeneous wave equation using a continuous time Galerkin method. We present fully implicit examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral, hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz preconditioning are used for solving the linear system. We introduce a time decomposition strategy in preconditioning which significantly improves performance when compared with unpreconditioned cases.Comment: 9 pages, 5 figures, 5 table

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Adaptive multilevel solvers for the discontinuous Petrov–Galerkin method with an emphasis on high-frequency wave propagation problems

This dissertation focuses on the development of fast and efficient solution schemes for the simulation of challenging problems in wave propagation phenomena. In particular, emphasis is given on high frequency acoustic and electromagnetic problems which are characterized by localized solutions. This kind of simulations are essential in various applications, such as ultrasonic testing, laser scanning and modeling of optical laser amplifiers. In wave simulations, the computational cost of any numerical method, is directly related to the frequency. In the high-frequency regime very fine meshes have to be used in order to satisfy the Nyquist criterion and overcome the pollution effect. This often leads to prohibitively expensive problems. Numerical methods based on standard Galerkin discretizations lack pre-asymptotic discrete stability and therefore adaptive mesh refinement strategies are usually inefficient. Additionally, the indefinite nature of the wave operator makes state of the art preconditioning techniques, such as multigrid, unreliable. In this work, a promising alternative approach is followed within the framework of the discontinuous Petrov–Galerkin (DPG) method. The DPG method offers numerous advantages for our problems of interest. First and foremost, it offers mesh and frequency independent discrete stability even in the pre-asymptotic region. This is made possible by computing, on the fly, an optimal test space as a function of the trial space. Secondly, it provides a built-in local error indicator that can be used to drive adaptive refinements. Combining these two properties together, reliable adaptive refinement strategies are possible which can be initiated from very coarse meshes. Lastly, the DPG method can be viewed as a minimum residual method, and therefore it always delivers symmetric (Hermitian) positive definite stiffness matrix. This is a desirable advantage when it comes to the design of iterative solution algorithms. Conjugate Gradient based solvers can be employed which can be accelerated by domain decomposition (one- or multi- level) preconditioners for symmetric positive definite systems. Driven by the aforementioned properties of the DPG method, an adaptive multigrid preconditioning technology is developed that is applicable for a wide range of boundary value problems. Unlike standard multigrid techniques, our preconditioner involves trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. Integration of the iterative solver within the DPG adaptive procedure turns out to be crucial in the simulation of high frequency wave problems. A collection of numerical experiments for the solution of linear acoustics and Maxwell equations demonstrate the efficiency of this technology, where under certain circumstances uniform convergence with respect to the mesh size, the polynomial order and the frequency can be achieved. The construction is complemented with theoretical estimates for the condition number in the one-level setting.Computational Science, Engineering, and Mathematic