arXiv subjects

Andreas Dedner

Publications and source records attributed to Andreas Dedner.

At least 19 recordsLinked to original sources

arXiv2026

Virtual element methods for a class of fully nonlinear elliptic PDEs

We study virtual element discretizations of a well-known variational formulation in $H^2$ of Hamilton-Jacobi-Bellman and Isaacs equations with Cordes coefficients. We show that the use of $H^2$ conforming virtual element spaces leads to a relatively simple analysis, bypassing the need for discrete Miranda-Talenti estimates that typically arises when using nonconforming schemes. We investigate how the polynomial degree of projection operators, especially for lower order terms, affects both the error analysis and robustness of the proposed schemes. We also show the possibility of imposing weakly the Dirichlet boundary condition, which simplifies the implementation of the method in some virtual element codes. Our results are complemented by numerical experiments in which we compare the convergence of different variants of the scheme for some test problems.

math.NA↗

arXiv2025

A posteriori analysis for nonlinear convection-diffusion systems

This work provides reliable a posteriori error estimates for Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems. The classes of systems we study are quite general with a focus on convection-dominated and degenerate parabolic problems. Our a posteriori error bounds are valid for a family of discontinuous Galerkin spatial discretizations and various temporal discretizations that include explicit and implicit-explicit time-stepping schemes, popular tools for practical simulations of this class of problem. We prove that our estimators provide reliable upper bounds for the error of the numerical method and present numerical evidence showing that they achieve the same order of convergence as the error. Since one of our main interests is the convection dominant case, we also track the dependence of the estimator on the viscosity coefficient.

math.NA↗

arXiv2025

Diffuse Domain Methods with Dirichlet Boundary Conditions

The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates the problem on a larger, simple domain where the complex geometry is represented by a smooth phase-field function. This paper introduces and analyses several new DDM methods for solving problems with Dirichlet boundary conditions. We derive two new methods from the mixed formulation of the governing equations. This approach transforms the essential Dirichlet conditions into natural boundary conditions. Additionally, we develop coercive formulations based on Nitsche's method, and provide proofs of coercivity for all new and key existing approximations. Numerical experiments demonstrate the improved accuracy of the new methods, and reveal the balance between $L^2$ and $H^1$ errors. The practical effectiveness of this approach is demonstrated through the simulation of the incompressible Navier-Stokes equations on a benchmark fluid dynamics problems.

math.NA↗

arXiv2025

Bryne: sustainable prototyping of finite element models

Open-source simulation frameworks are evolving rapidly to provide accessible tools for the numerical solution of partial differential equations. Modern finite element (FEM) software such as FEniCS, Firedrake, or dune-fem alleviates the need for modelers to recode the discretization and linear solver backend for each application and enables rapid prototyping of solvers. However, while it has become easier to build prototype FEM models, creating a solver reusable beyond its specific initial simulation setup remains difficult. Moreover, simulation setups typically cover an ample input parameter space, and tracking complex metadata on research project time scales has become a challenge. This implies the need to supplement model development with a coding-intensive complementary workstream, seldom developed for sustainable reuse. To address these issues, we introduce our open-source Python package Bryne. Bryne is an object-oriented framework for FEM solvers built with the dune-fem Python API. In this article, we describe how it helps to evolve rapid-prototyping solver development into sustainable simulation building. First, we show how to translate a minimal dune-fem solver into a Bryne FEM model to build human-readable, metadata-enriched simulations. Bryne then offers a simulation driver and model coupling interfaces to combine implemented solvers in operator-split multiphysics simulations. The resulting reproducibility-enabled infrastructure allows users to tackle complex simulation setups without sacrificing backend flexibility. We demonstrate the workflow on a convection-coupled phase-change simulation, where a discontinuous Galerkin flow solver is coupled with a solver for solidification phase change.

physics.comp-ph↗

arXiv2025

DDFEM: A Python Package for Diffuse Domain Methods

Solving partial differential equations (PDEs) on complex domains can present significant computational challenges. The Diffuse Domain Method (DDM) is an alternative that reformulates the partial differential equations on a larger, simpler domain. The original geometry is embedded into the problem by representing it with a phase-field function. This paper introduces ddfem, an extensible Python package to provide a framework for transforming PDEs into a Diffuse Domain formulation. We aim to make the application of a variety of different Diffuse Domain approaches more accessible and straightforward to use. The ddfem package includes features to intuitively define complex domains by combining signed distance functions and provides a number of DDM transformers for general second evolution equations. In addition, we present a new approach for combining multiple boundary conditions of different types on distinct boundary segments. This is achieved by applying a normalised weighting, derived from multiple phase fields, to combine the additional boundary terms in the Diffuse Domain formulations. The domain definition and Diffuse Domain transformation provided by our package are designed to be compatible with a wide range of existing finite element solvers without requiring code alterations. Both the original (non-linear) PDEs provided by the user and the resulting transformed PDEs on the extended domain are defined using the Unified Form Language UFL which is a domain specific language used by a number of software packages. Our experiments were carried out using the Dune-Fem framework.

math.NA↗

arXiv2025

The Distributed and Unified Numerics Environment (DUNE), Version 2.10

Version 2.10 of the Distributed and Unified Numerics Environment DUNE introduces a range of enhancements across its core and extension modules, with a continued emphasis on modern C++ integration and improved usability. This release extends support for C++20 features, particularly concepts, through comprehensive refinements in dune-common and dune-grid, enabling safer and more expressive generic programming paradigms. A notable advancement is the improved support for curved geometries, including new geometry implementations and a more flexible interface. Data structures have been modernized through native support for std::mdspan and std::mdarray, performance improvements in sparse matrices, and tools for visualization of matrix patterns. The build system has been restructured towards a modern CMake workflow, emphasizing target-based configuration and improved automation. Furthermore, new local finite elements have been introduced to broaden numerical capabilities. The release also brings updates across DUNE extensions, as well as improvements to infrastructure and module-level components.

cs.MS↗

arXiv2025

Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods applied to Numerical Weather Prediction

Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in $z$-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. %The lack of efficient solvers suitable for contemporary computer architectures inhibits wider adoption of DG methods. This motivates our research to construct a Jacobian-free precondtioner for high order DG discretizations. The preconditioner is based on a multigrid method constructed for a low order finite volume discretization defined on a subgrid of the DG mesh. We design a computationally efficient and mass conservative mapping between the grids. As smoothers, explicit Runge-Kutta pseudo time iterations are used, which can be implemented in parallel in a Jacobian-free low-memory manner. We consider DG Methods for the Euler equations and for viscous flow equations in 2D, both with gravity, in a well balanced formulation. Numerical experiments in the software framework DUNE-FEM on atmospheric flow problems show the benefit of this approach.

math.NA↗

arXiv2024

Convergence Properties of Iteratively Coupled Surface-Subsurface Models

Surface-subsurface flow models for hydrological applications solve a coupled multiphysics problem. This usually consists of some form of the Richards and shallow water equations. A typical setup couples these two nonlinear partial differential equations in a partitioned approach via boundary conditions. Full interaction between the subsolvers is ensured by an iterative coupling procedure. This can be accelerated using relaxation. In this paper, we apply continuous and fully discrete linear analysis techniques to study an idealized, linear, 1D-0D version of a surface-subsurface model. These result in explicit expressions for the convergence factor and an optimal relaxation parameter, depending on material and discretization parameters. We test our analysis results numerically for fully nonlinear 2D-1D experiments based on existing benchmark problems. The linear analysis can explain fast convergence of iterations observed in practice for different materials and test cases, even though we are not able to capture various nonlinear effects.

math.NA↗

arXiv2024

Isoparametric Virtual Element Methods

We present two approaches to constructing isoparametric Virtual Element Methods of arbitrary order for linear elliptic partial differential equations on general two-dimensional domains. The first method approximates the variational problem transformed onto a computational reference domain. The second method computes a virtual domain and uses bespoke polynomial approximation operators to construct a computable method. Both methods are shown to converge optimally, a behaviour confirmed in practice for the solution of problems posed on curved domains.

math.NA↗

arXiv2024

On the stabilization of a virtual element method for an acoustic vibration problem

In this paper we introduce an abstract setting for the convergence analysis of the virtual element approximation of an acoustic vibration problem. We discuss the effect of the stabilization parameters and remark that in some cases it is possible to achieve optimal convergence without the need of any stabilization. This statement is rigorously proved for lowest order triangular element and supported by several numerical experiments.

math.NA↗

arXiv2022

A framework for implementing general virtual element spaces

In this paper we present a framework for the construction and implementation of general virtual element spaces based on projections built from constrained least squares problems. Building on the triples used for finite element spaces, we introduce the concept of a VEM tuple which encodes the necessary building blocks to construct these projections. Using this approach, a wide range of virtual element spaces can be defined. We discuss $H^k$-conforming spaces for $k=1,2$ as well as divergence and curl free spaces. This general framework has the advantage of being easily integrated into any existing finite element package and we demonstrate this within the open source software package DUNE.

math.NA↗

arXiv2021

A higher order nonconforming virtual element method for the Cahn-Hilliard equation

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 semidiscrete (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.

math.NA↗

arXiv2020

Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Method

This paper discusses a Python interface for the recently published DUNE-FEM-DG module which provides highly efficient implementations of the Discontinuous Galerkin (DG) method for solving a wide range of non linear partial differential equations (PDE). Although the C++ interfaces of DUNE-FEM-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work easier user interfaces based on Python and the Unified Form Language are provided to open DUNE-FEM-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and first order hyperbolic PDEs.

cs.MS↗

arXiv2020

Robust nonconforming virtual element methods for general fourth order problems with varying coefficients

We present a class of nonconforming virtual element methods for general fourth order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth order problems with varying coefficients. We also discuss fourth order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth order problems.

math.NA↗

arXiv2020

A surface finite element method for computational modelling of cell blebbing

Cell blebs are protrusions of the cell membrane and can be instrumental for cell migration. We derive a continuum model for the mechanical and geometrical aspects of the onset of blebbing in terms of a force balance. It is abstract and flexible in that it allows for amending force contributions related to membrane tension or the presence of linker molecules between membrane and cell cortex. The deforming membrane and all forces are expressed by means of a parametrisation over a stationary reference surface. A variational formulation is presented and analysed for well-posedness. For this purpose, we derive a semi-discrete scheme based on the surface finite element method. We provide a convergence result and estimates of the error due to the spatial discretisation. Furthermore, we present a computational framework where specific models can be implemented and later on conveniently amended if desired, using a domain specific language implemented in Python. While the high level program control can be done within the Python scripting environment, the actual computationally expensive step of evolving the solution over time is carried out by binding to an efficient software backend. Cell membrane geometries given in terms of a parametrisation or obtained from image data can be accounted for. A couple of numerical simulation results illustrate the approach.

math.NA↗

arXiv2019

The DUNE Framework: Basic Concepts and Recent Developments

This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first Dune version in 2007 and the main papers describing that state [1, 2]. This discussion is accompanied with a description of various advanced features, such as coupling of domains and cut cells, grid modifications such as adaptation and moving domains, high order discretizations and node level performance, non-smooth multigrid methods, and multiscale methods. A brief discussion on current and future development directions of the framework concludes the paper.

cs.MS↗

arXiv2019

Residual estimates for post-processors in elliptic problems

In 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 of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Enhancing filter and Superconvergent Patch Recovery. Extensive numerical tests are conducted that confirm our analytic findings.

math.NA↗

arXiv2018

The Dune Python Module

In this paper we present the new Dune-Python module which provides Python bindings for the Dune core, which is a C++ environment for solving partial differential equations. The aim of this new module is to firstly provide the general infrastructure for exporting realizations of statically polymorphic interfaces based on just-in-time compilation and secondly to provide bindings for the central interfaces of the dune core modules. In the first release we focus on the grid interface. Our aim is to only introduce a thin layer when passing objects into Python which can be removed when the object is passed back into a C++ algorithm. Thus no efficiency is lost and little additional code maintenance cost is incurred. To make the transition for Dune users to the Python environment straightforward the Python classes provide a very similar interface to their C++ counterparts. In addition, vectorized versions of many interfaces allow for more efficient code on the Python side. The infrastructure for exporting these interfaces and the resulting bindings for a Dune grid are explained in detail in this paper for both experienced Dune users and others interested in a flexible Python environment for implementing grid based schemes for solving partial differential equations.

cs.MS↗