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This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element…
In this paper, we present explicit expressions for conforming finite element function spaces, basis functions, and degrees of freedom on the pentatope and tetrahedral prism elements. More generally, our objective is to construct finite…
Matrix Riccati differential equations arise in many different areas and are particular important within the field of control theory. In this paper we consider numerical integration for large-scale systems of stiff matrix Riccati…
We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we…
We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution…
Numerical evaluations of Feynman integrals often proceed via a deformation of the integration contour into the complex plane. While valid contours are easy to construct, the numerical precision for a multi-loop integral can depend…
We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE…
We present new finite elements for solving the Riesz maps of the de Rham complex on triangular and tetrahedral meshes at high order. The finite elements discretize the same spaces as usual, but with different basis functions, so that the…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
This article addresses the research question if and how the finite cell method, an embedded domain finite element method of high order, may be used in the simulation of metal deposition to harvest its computational efficiency. This…
In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient and favorably compares with other…
In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this…
In this paper, we present a NURBS-enhanced finite element method that integrates the NURBS-based boundary representation of a geometric domain into a standard finite element framework for hexahedral meshes. We decompose an open, bounded,…
The scaled boundary finite element method is known for its capability in reproducing highly-detailed solution fields. This, however, is only attainable in those cases where analytical solutions exist. Many others invoke the use of numerical…
In this paper, we describe a numerical approach to evaluate Feynman loop integrals. In this approach the key technique is a combination of a numerical integration method and a numerical extrapolation method. Since the computation is carried…
We propose a general algorithm for non-conforming adaptive mesh refinement (AMR) of unstructured meshes in high-order finite element codes. Our focus is on h-refinement with a fixed polynomial order. The algorithm handles triangular,…
We consider the problem of numerically integrating functions with hyperplane discontinuities over the entire Euclidean space in many dimensions. We describe a simple process through which the Euclidean space is partitioned into simplices on…
We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be…
Trimming is a ubiquitous operation in computer-aided-design whereby parts of a geometry are merged, intersected, or simply discarded. While it grants virtually unlimited flexibility in geometric design, it introduces a plethora of other…
The efficient numerical integration of large-scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications. Standard numerical methods applied to such problems require an unduly…