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### The Dune Python Module [article]

Andreas Dedner, Martin Nolte
2018 arXiv   pre-print
CommOp .set , data) c by the authors, 2018 Dedner, Nolte  ...  At the time of writing we are developing the Dune-Fempy module Connellan et al. [2018] exporing the interfaces in Dune-Fem Dedner et al. [2010] to Python.  ...

### Discontinuous Galerkin methods for nonvariational problems [article]

Andreas Dedner, Tristan Pryer
2013 arXiv   pre-print
We extend the finite element method introduced by Lakkis and Pryer [2011] to approximate the solution of second order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the NVFEM as a mixed method whereby the finite element Hessian is an auxiliary variable in the formulation. Representing the finite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the
more » ... compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble, Thus this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unified framework set out in Arnold et. al. [2001]. We also give an apriori analysis of the method. The analysis applies to any consistent representation of the finite element Hessian, thus is applicable to the previous works making use of continuous Galerkin approximations.

### Adaptive discontinuous Galerkin methods on surfaces [article]

2014 arXiv   pre-print
The bilinear form A IP Γ is related to the original problem (P Γ ) in the following way: Dedner et al. (2013) .  ...  Provided that the penalty parameters ω e h are large enough, boundedness and stability of A IP Γ in H 1 (Γ ) +V l h follow from Lemma 3.4 in Dedner et al. (2013) .  ...

### Phase field methods for binary recovery [article]

Charles Brett, Andreas S. Dedner, Charles M. Elliott
2013 arXiv   pre-print
We consider the inverse problem of recovering a binary function from blurred and noisy data. Such problems arise in many applications, for example image processing and optimal control of PDEs. Our formulation is based on the Mumford-Shah model, but with a phase field approximation to the perimeter regularisation. We use a double obstacle potential as well as a smooth double well potential. We introduce an iterative method for solving the problem, develop a suitable discretisation of this
more » ... ve method, and prove some convergence results. Numerical simulations are presented which illustrate the usefulness of the approach and the relative merits of the phase field models.

### Discontinuous Galerkin methods for hyperbolic and advection-dominated problems on surfaces [article]

2015 arXiv   pre-print
This method was first considered in Dedner et al. (2013) for the elliptic problem including a mass term of the form εuv.  ...  The proof of the first estimate follows similar arguments to that of Lemma 3.3 in Dedner et al. (2013) . The second estimate is given in (2.17) in Demlow (2009).  ...

### A surface finite element method for computational modelling of cell blebbing [article]

Björn Stinner, Andreas Dedner, Adam Nixon
2020 arXiv   pre-print
Andreas Dedner was core developer of the Python bindings and of the DUNE software framework and set parts of the paper.  ...

### A higher order nonconforming virtual element method for the Cahn-Hilliard equation [article]

Andreas Dedner, Alice Hodson
2021 arXiv   pre-print
In this paper we develop a fully nonconforming virtual element method (VEM) of arbitrary approximation order for the two dimensional Cahn-Hilliard equation. We carry out the error analysis for the continuous-in-time scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.

### The DUNE-ALUGrid Module [article]

Martin Alkämper, Andreas Dedner, Robert Klöfkorn, Martin Nolte
2015 arXiv   pre-print
In this paper we present the new DUNE-ALUGrid module. This module contains a major overhaul of the sources from the ALUgrid library and the binding to the DUNE software framework. The main changes include user defined load balancing, parallel grid construction, and an redesign of the 2d grid which can now also be used for parallel computations. In addition many improvements have been introduced into the code to increase the parallel efficiency and to decrease the memory footprint. The original
more » ... LUGrid library is widely used within the DUNE community due to its good parallel performance for problems requiring local adaptivity and dynamic load balancing. Therefore, this new model will benefit a number of DUNE users. In addition we have added features to increase the range of problems for which the grid manager can be used, for example, introducing a 3d tetrahedral grid using a parallel newest vertex bisection algorithm for conforming grid refinement. In this paper we will discuss the new features, extensions to the DUNE interface, and explain for various examples how the code is used in parallel environments.

### Formulation, Implementation and Validation of the Horizontal Coupling Method for 1D/2D Shallow Water Flow Models [article]

Chinedu Nwaigwe, Andreas Dedner
2017 arXiv   pre-print
One dimensional (1D) simulations of the flow and flooding of open channels are known to be inaccurate as the flow is multi-dimensional in nature, especially at the flooded regions. However, multi-dimensional simulations, even in two dimensions (2D), are computationally expensive, hence the problem of efficiently coupling 2D and 1D simulations for the flow and flooding of open channels has been the subject of much research and is investigated in this paper. We adopt a 1D model with coupling term
more » ... for the channel flow and the 2D shallow water flow model for the floodplain. The 1D model with coupling term is derived by integrating the 3D Free Surface Euler equations but without imposing any restriction on the channel width variations. term. Finite volume methods are formulated for both the 2D and 1D models including a discrete coupling term in closed form. Coupling is achieved through the discrete coupling term in the 1D model and the lateral numerical fluxes in the 2D model. Since the lateral discharge in the channel cannot be guaranteed to be zero during flooding, we aim to recover the lateral variation by computing two lateral discharges over each cross section and propose to use an ad-hoc model based on the y-discharge equation in the 2D model for this purpose. We then propose the numerical scheme for this ad-hoc model following the hydrostatic reconstruction philosophy. Then, we show that the resulting method, named Horizontal Coupling Method (HCM), is well-balanced; we introduce the no-numerical flooding property and also show that the method satisfies the property. Three numerical test cases are used to verify the performance of the method. The results show that the method performs well in both accuracy and efficiency and also approximates the channel lateral discharges with very good accuracy and little computational overhead.

### Convection in a Single Column -- Modelling, Algorithm and Analysis [article]

Onno Bokhove, Bin Cheng, Andreas Dedner, Gavin Esler, John Norbury, Matthew R. Turner, Jacques Vanneste, Mike Cullen
2016 arXiv   pre-print
The group focused on a model problem of idealised moist air convection in a single column of atmosphere. Height, temperature and moisture variables were chosen to simplify the mathematical representation (along the lines of the Boussinesq approximation in a height variable defined in terms of pressure). This allowed exact simple solutions of the numerical and partial differential equation problems to be found. By examining these, we identify column behaviour, stability issues and explore the feasibility of a more general solution process.

### Residual Estimates for Post-processors in Elliptic Problems

Andreas Dedner, Jan Giesselmann, Tristan Pryer, Jennifer K Ryan
2021 Journal of Scientific Computing
AbstractIn this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that "tweaks" a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes
more » ... f very popular reconstruction operators, the Smoothness-Increasing Accuracy-Conserving filter and superconvergent patch recovery. Extensive numerical tests are conducted that confirm our analytic findings.

### Discontinuous Galerkin Methods for a Class of Nonvariational Problems

Andreas Dedner, Tristan Pryer
2021 Communications on Applied Mathematics and Computation
AbstractWe extend the finite element method introduced by Lakkis and Pryer (SIAM J. Sci. Comput. 33(2): 786–801, 2011) to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the "finite element Hessian" as an auxiliary variable in the formulation. Representing the finite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity
more » ... ern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble; thus, this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unified framework set out in Arnold et al. (SIAM J. Numer. Anal. 39(5): 1749–1779, 2001/2002). We also give an a posteriori analysis of the method in the case where the problem has a strong solution. The analysis applies to any consistent representation of the finite element Hessian, and thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is presented showing that the method works well also in a more general setting.

### Python Framework for HP Adaptive Discontinuous Galerkin Method for Two Phase Flow in Porous Media [article]

Andreas Dedner, Birane Kane, Robert Klöfkorn, Martin Nolte
2018 arXiv   pre-print
In this paper we present a framework for solving two phase flow problems in porous media. The discretization is based on a Discontinuous Galerkin method and includes local grid adaptivity and local choice of polynomial degree. The method is implemented using the new Python frontend Dune-FemPy to the open source framework Dune. The code used for the simulations is made available as Jupyter notebook and can be used through a Docker container. We present a number of time stepping approaches
more » ... from a classical IMPES method to fully coupled implicit scheme. The implementation of the discretization is very flexible allowing for test different formulations of the two phase flow model and adaptation strategies.

### Dune-MMesh: The Dune Grid Module for Moving Interfaces

Samuel Burbulla, Andreas Dedner, Maximilian Hörl, Christian Rohde
2022 Journal of Open Source Software
Various examples in Python have been implemented based on the discretization module Dune-Fem (Dedner et al., 2020) that demonstrate the versatile applicability of Dune-MMesh.  ...

### Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio [article]

Andreas Dedner, Eike Hermann Müller, Robert Scheichl
2015 arXiv   pre-print
Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods which are
more » ... y popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the analysis to three dimensions under slightly weakened assumptions, and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment (DUNE), which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR supercomputer.
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