Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

WAVEx: Stabilized Finite Elements for Spectral Wind Wave Models Using FEniCSx

Several potential FEM discretizations of the Wave Action Balance Equation are discussed. The methods, which include streamline upwind Petrov-Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented using the open source finite element library FEniCSx for simplified 2-D cases. Open source finite element libraries, such as FEniCSx, typically only support geometries up to dimension of 3. The Wave Action Balance Equation is 4 dimensions in space so this presents difficulties. A method to use a FEM library, such as FEniCSx, to solve problems in domains with dimension larger than 4 using the product basis is discussed. A new spectral wind wave model, WAVEx, is formulated and implemented using the new finite element library FEniCSx. WAVEx is designed to allow for construction of multiple FEM discretizations with relatively small modifications in the Python code base. An example implementation is then demonstrated with WAVEx using continuous finite elements and SUPG stabilization in geographic/spectral space. For propagation in time, a generalized one step implicit finite difference method is used. When source terms are active, the second order operator splitting scheme known as Strang splitting is used. In the splitting scheme, propagation is solved using the aforementioned implicit method and the nonlinear source terms are treated explicitly using second order Runge-Kutta. Several test cases which are part of the Office for Naval Research Test Bed (ONR Test Bed) are demonstrated both with and without 3rd generation source terms and results are compared to analytic solutions, observations, and SWAN output

A controllability method for Maxwell's equations

We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached

Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems

In this work, we consider space-time goal-oriented a posteriori error estimation for parabolic problems. Temporal and spatial discretizations are based on Galerkin finite elements of continuous and discontinuous type. The main objectives are the development and analysis of space-time estimators, in which the localization is based on a weak form employing a partition-of-unity. The resulting error indicators are used for temporal and spatial adaptivity. Our developments are substantiated with several numerical examples.Comment: Changes in v2: - Updated the title - Reworked space-time function spaces - Added cG(1) in time partition-of-unity - Added links to the now published codes used for this work - Added further reference

A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow

We present a comparison between hybridized and non-hybridized discontinuous Galerkin methods in the context of target-based hp-adaptation for compressible flow problems. The aim is to provide a critical assessment of the computational efficiency of hybridized DG methods. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. Using a discrete-adjoint approach, sensitivities with respect to output functionals are computed to drive the adaptation. From the error distribution given by the adjoint-based error estimator, h- or p-refinement is chosen based on the smoothness of the solution which can be quantified by properly-chosen smoothness indicators. Numerical results are shown for subsonic, transonic, and supersonic flow around the NACA0012 airfoil. hp-adaptation proves to be superior to pure h-adaptation if discontinuous or singular flow features are involved. In all cases, a higher polynomial degree turns out to be beneficial. We show that for polynomial degree of approximation p=2 and higher, and for a broad range of test cases, HDG performs better than DG in terms of runtime and memory requirements