This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Benchmark configurations for quantitative validation and comparison of incompressible interfacial flow codes, which model two-dimensional bubbles rising in liquid columns, are proposed. The benchmark quantities: circularity, center of mass, and mean rise velocity are defined and measured to monitor convergence toward a reference solution. Comprehensive studies are undertaken by three independent research groups, two representing Eulerian level set finite-element codes and one representing an arbitrary Lagrangian-Eulerian moving grid approach.

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Quantitative benchmark computations of two-dimensional bubble dynamics

Effects of large computational time steps on the computed turbulence were investigated using a fully implicit method. In turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined. Turbulence fluctuations could not be sustained if the computational time step was near or larger than the Kolmogorov time scale.

Effects of the computational time step on numerical solutions for turbulent flow

Notes on numerical fluid mechanics

The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $H(div)$-conforming finite ...

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

https://www.wias-berlin.de/people/john/ELECTRONIC_PAPERS/JK07.CMAME.pdf

On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review

We present a numerical study of a two-dimensional time-dependent flow around a cylinder. Its main objective is to provide accurate reference values for the maximal drag and lift coefficient at the cylinder and for the pressure difference between the front and the back of the cylinder at the final time. In addition, the accuracy of these values obtained with different time stepping schemes and different finite element methods is studied

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Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder

https://publikationen.sulb.uni-saarland.de/bitstream/20.500.11880/26558/1/preprint_219_08.pdf

On Finite Element Methods for 3D Time-Dependent Convection-Diffusion-Reaction Equations with Small Diffusion

Techniques for the reconstruction of a distribution from a finite number of its moments

This paper presents a variational multiscale method (VMS) for the incompressible Navier--Stokes equations which is defined by a large scale space LH for the velocity deformation tensor and a turbulent viscosity $\nu_T$. The connection of this method to the standard formulation of a VMS is explained. The conditions on LH under which the VMS can be implemented easily and efficiently into an existing finite element code for solving the Navier--Stokes equations are studied. Numerical tests with the Smagorinsky large eddy simulation model for $\nu_T$ are presented.

Volker John

Papers

On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review

Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder

On Finite Element Methods for 3D Time-Dependent Convection-Diffusion-Reaction Equations with Small Diffusion

Techniques for the reconstruction of a distribution from a finite number of its moments

A Finite Element Variational Multiscale Method for the Navier-Stokes Equations