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MooAFEM: An object oriented Matlab code for higher-order adaptive FEM for (nonlinear) elliptic PDEs

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Abstract

We present an easily accessible, object oriented code (written exclusively in Matlab) for adaptive finite element simulations in 2D. It features various refinement routines for triangular meshes as well as fully vectorized FEM ansatz spaces of arbitrary polynomial order and allows for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. Due to the object oriented programming paradigm, the code can be used easily and is readily extensible. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.

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... Rearranging this estimate, we conclude the proof of (38). It remains to verify ( ) < ∞ for some > 0. Note that (37) guarantees that 0 ≤ ≤ 2 −1 ≤ 2 0 for all ∈ ℕ. ...
... The following numerical experiments employ the Matlab software package MooAFEM from [38]. 1 The first numerical example illustrates the performance of Algorithm B for a symmetric linear elliptic PDE with a strong jump in the diffusion coefficient and compares Algorithm B to the Algorithm A with exact solution. The second numerical example demonstrates the efficiency of Algorithm C for a nonsymmetric general second-order linear elliptic PDE with a moderate convection. ...
Article
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The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computation time. To this end, algorithmically, the standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This work tackles several shortcomings of previous approaches by introducing a new proof strategy. Previously, the analysis of the algorithm required several parameters to be fine-tuned. This work leaves the classical reasoning and introduces a summability criterion for R-linear convergence to remove restrictions on those parameters. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from Feischl (2022) [22]. Importantly, this paves the way towards extending the analysis of AFEM with inexact solver to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity parameters.
... Using the open-source object-oriented 2D Matlab code MooAFEM [22], we present a detailed numerical study of both the algebraic solver and the adaptive algorithm, including higher-order experiments and jumping coefficients. ...
... This section investigates the numerical performance of the proposed multigrid solver of Algorithm 2.1 and the adaptive Algorithm 3.1. The Matlab implementation of the multigrid solver is embedded into the MooAFEM 1 framework from [22]. Throughout, we choose the marking parameter = 0.5 in the adaptive Algorithm 3.1 and = (0, 0) ⊤ . ...
Article
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In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p1p \ge 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.
... Furthermore, we present a detailed numerical study of both the algebraic solver and the full adaptive algorithm, including higher-order experiments and jumping coefficients. The experiments are implemented in the open-source object oriented 2D Matlab code MooAFEM [IP22]. ...
... This section investigates the numerical performance of the proposed multigrid solver of Algorithm B and the adaptive Algorithm A. The Matlab implementation of the multigrid solver is embedded into the MooAFEM 1 framework from [IP22]. Throughout, we choose the marking parameter θ = 0.5 in the adaptive Algorithm A. We introduce the following test case: ...
Preprint
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In this work, a fully adaptive finite element algorithm for symmetric second-order elliptic diffusion problems with inexact solver is developed. The discrete systems are treated by a local higher-order geometric multigrid method extending the approach of [Mira\c{c}i, Pape\v{z}, Vohral\'{i}k, SIAM J. Sci. Comput. (2021)]. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p1p \ge 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator is h- and p-robustly equivalent to the algebraic error. The proofs rely on suitably chosen robust stable decompositions and a strengthened Cauchy-Schwarz inequality on bisection-generated meshes. Together, this yields that the proposed adaptive algorithm has optimal computational cost. Numerical experiments confirm the theoretical findings.
... In this section, we present numerical experiments using the open source software package MooAFEM [51]. 1 In the following, Steps (I) and (II) of Algorithm 1 employ the optimal hp-robust local multigrid method from [32] as an algebraic solver. If not explicitly stated otherwise, we choose the parameters = 0.5, = 0.5, sym = alg = 0.7 ...
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We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest G(u^\star) with a linear goal functional G and the solution u^\star to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for non-symmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAIS-FEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.
... There are various theories to determine the behaviour of plates at different boundary conditions. Classical plate theory (CPT) [22], first-order shear deformation theory (FSDT) [23,24], higher-order deformation theory (HSDT) [25], third-order shear deformation theory (TSDT) [26], sinusoidal shear deformation theory (SSDT) [27] and tangential shear deformation theory (TSDT) [28], finite element method [29,30] can be given as examples of these theories. Different theories have been developed to determine the complex deformation behaviour more accurately by considering additional terms in the displacement field. ...
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Purpose In this study, the effect of foam structure on the thermomechanical behaviour of high void ratio porous FGM piezoelectric smart nanoplates is investigated. Method The material of the smart nanoplate consists of PZT-4 on the bottom surface and BaTiO 3 on the top surface and is formed by functional grading of these two materials along the thickness of the plate. Four different foam distribution models are modelled to examine the foam structure of the highly porous smart nanoplate, which has become widespread in biosensor applications. For this reason, uniform, symmetrical, top symmetrical and bottom symmetrical foam distribution models are created up to 75% void ratio. To determine the nano size, equations of motion are obtained by using nonlocal strain gradient elasticity and sinusoidal shear deformation theories together, and these equations are solved by the Navier method according to general boundary conditions. Result and Conclusions As a result of the analysis, it is observed that the applied external electric potential creates a softening effect on the plates with the piezoelectric elasticity effect and therefore reduces the thermal buckling temperatures. It is observed that the presence of the foam structure significantly improves the thermal resistance of the material and increases the buckling temperatures. It is also observed that the foam distribution model has significant effects on the thermomechanical behaviour.
... For higher order PDEs, a finite element method (FEM) may be conveniently used for PDE solving and can give an accurate solution using extensive computational resources (Innerberger and Praetorius 2023). Also, the multi-iteration solution limits practicality. ...
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Mathematics lies at the heart of engineering science and is very important for capturing and modeling of diverse processes. These processes may be naturally-occurring or man-made. One important engineering problem in this regard is the modeling of advanced mathematical problems and their analysis. Partial differential equations (PDEs) are important and useful tools to this end. However, solving complex PDEs for advanced problems requires extensive computational resources and complex techniques. Neural networks provide a way to solve complex PDEs reliably. In this regard, large-data models are new generation of techniques, which have large dependency capturing capabilities. Hence, they can richly model and accurately solve such complex PDEs. Some common large-data models include Convolutional neural networks (CNNs) and their derivatives, transformers, etc. In this literature survey, the mathematical background is introduced. A gentle introduction to the area of solving PDEs using large-data models is given. Various state-of-the-art large-data models for solving PDEs are discussed. Also, the major issues and future scope of the area are identified. Through this literature survey, it is hoped that readers will gain an insight into the area of solving PDEs using large-data models and pursue future research in this interesting area.
... It provides increased flexibility and convergence properties compared to the "conventional" FEM. There are recent MATLAB implementations including triangular elements [12] and rectangular elements [7]. ...
... The experiments are performed with the open-source software package MooAFEM [IP23]. In the following, Algorithm A employs the optimal local hp-robust multigrid method [IMPS23] as algebraic solver. ...
Preprint
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We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AIL-FEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.
... In this section, we present numerical experiments using the open source software package MooAFEM [IP23] 1 . In the following, Step (I) and (II) of Algorithm A employ the optimal hp-robust local multigrid method from [IMPS22] as an algebraic solver. ...
Preprint
Full-text available
We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest G(u)G(u^\star) with a linear goal functional G and the solution uu^\star to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contrac-tive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.
... The following numerical experiments employ the Matlab software package MooAFEM from [IP23]. 1 The first numerical example illustrates the performance of Algorithm B for a symmetric linear elliptic PDE with a strong jump in the diffusion coefficient and compares Algorithm B to the Algorithm A with exact solution. The second numerical example demonstrates the efficiency of Algorithm C for a nonsymmetric general second-order linear elliptic PDE with a moderate convection. ...
Preprint
Full-text available
The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This work tackles several shortcomings of previous approaches by introducing a new proof strategy. First, the algorithm requires several fine-tuned parameters in order to make the underlying analysis work. A redesign of the standard line of reasoning and the introduction of a summability criterion for R-linear convergence allows us to remove restrictions on those parameters. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from [Feischl, Math. Comp., 91 (2022)]. Importantly, this paves the way towards extending the analysis to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity parameters.
... We consider the model problem ( Innerberger & Praetorius (2023). In the following, Algorithm A employs the optimal local hp-robust multigrid method from Innerberger et al. (2022) as algebraic solver and the standard residual error estimator η . ...
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We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.
... In this section, we test and illustrate Algorithm B with numerical experiments. All experiments were implemented using the Matlab code MooAFEM [32]. Throughout, Ω ⊂ R 2 and we use = ( 1 , 2 ) ∈ Ω to denote the Cartesian coordinates. ...
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We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element method (AILFEM) which steers the local mesh refinement as well as the iterative linearization of the arising nonlinear discrete equations. To this end, we employ a damped Zarantonello iteration so that, in each step of the algorithm, only a linear Poisson-type equation has to be solved. We prove that the proposed AILFEM strategy guarantees convergence with optimal rates, where rates are understood with respect to the overall computational complexity (i.e., the computational time). Moreover, we formulate and test an adaptive algorithm where also the damping parameter of the Zarantonello iteration is adaptively adjusted. Numerical experiments underline the theoretical findings.
... special data structures are needed [5,10]. A recent MATLAB contribution [8] provides an object-oriented approach to implement hp-FEM on triangles with adaptive h-refinement. ...
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... The number of samples is adjusted to balance statistical error and discretization bias on each level. The experiment has been implemented in MATLAB using the MooAFEM library [11] for the FE discretization. All arising linear systems are solved directly by the \-operator in MATLAB. ...
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... The Matlab implementation of the following experiments is embedded into the open source software package MooAFEM from [IP22]. In the following, Algorithm A employs the optimal local hp-robust multigrid method from [IMPS22] as algebraic solver. ...
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... In this section, we test and illustrate Algorithm B with numerical experiments. All experiments were implemented using the Matlab code MooAFEM [IP23]. Throughout, Ω ⊂ R 2 and we use x = (x 1 , x 2 ) ∈ Ω to denote the Cartesian coordinates. ...
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Full-text available
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... Throughout, we use newest vertex bisection (NVB) for mesh-refinement; see, e.g., [BDD04;Ste08]. All numerical examples are implemented using the Matlab AFEM package MooAFEM [IP22]. ...
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we construct a converging adaptive algorithm for linear elements applied to Poisson's equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriori error estimators to get a sequence of refined triangulations and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extensions to higher-order elements in two space dimensions and numerical results are included.
Book
Self-adaptive discretization methods nowadays are an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. In this monograph we review the most frequently used a posteriori error estimation techniques and apply them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. Our main goal is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The intention here is to present the basic principles using a minimal amount of notation and techniques. Chapters 4–6, on the other hand, are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.
Book
I Theoretical Foundations.- 1 Finite Element Interpolation.- 2 Approximation in Banach Spaces by Galerkin Methods.- II Approximation of PDEs.- 3 Coercive Problems.- 4 Mixed Problems.- 5 First-Order PDEs.- 6 Time-Dependent Problems.- III Implementation.- 7 Data Structuring and Mesh Generation.- 8 Quadratures, Assembling, and Storage.- 9 Linear Algebra.- 10 A Posteriori Error Estimates and Adaptive Meshes.- IV Appendices.- A Banach and Hilbert Spaces.- A.1 Basic Definitions and Results.- A.2 Bijective Banach Operators.- B Functional Analysis.- B.1 Lebesgue and Lipschitz Spaces.- B.2 Distributions.- B.3 Sobolev Spaces.- Nomenclature.- References.- Author Index.
Article
An application of the penalty method to the finite element method is analyzed. For a model Poisson equation with homogeneous Dirichlet boundary conditions, a variational principle with penalty is discussed. This principle leads to the solution of the Poisson equation by using functions that do not satisfy the boundary condition. The rate of convergence is discussed.
Article
In this paper an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever, for some s > 0, the solution can be approximated within a tolerance ε > 0 in energy norm by a continuous piecewise linear function on some partition with O(ε-1/s) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from [SINUM, 38 (2000), pp. 466-488] by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in several respects.
Article
The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data uD by functions uD,h in the trace space of a finite element space on D. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of uD,h, namely the nodal interpolation and the orthogonal projection in L2(D) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H1 and L2 a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.
Article
Algorithms are presented that enable the element matrices for the standard finite element space, consisting of continuous piecewise polynomials of degree n on simplicial elements in Rd, to be computed in optimal complexity Ω(n 2d). The algorithms (i) take into account numerical quadrature; (ii) are applicable to nonlinear problems; and (iii) do not rely on precomputed arrays containing values of one-dimensional basis functions at quadrature points (although these can be used if desired). The elements are based on Bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements.
Article
An adaptive finite element method is analyzed for approximating functionals of the solution of symmetric elliptic second order boundary value problems. We show that the method converges and derive a favorable upper bound for its convergence rate and computational complexity. We illustrate our theoretical findings with numerical results.
Book
Design patterns are a form of documentation that proposes solutions to recurring object-oriented software design problems. Design patterns became popular in software engineering thanks to the book published in 1995 by the Gang of Four (GoF): Erich Gamma, Richard Helm, Ralph Johnson, and John Vlissides. Since the publication of the book Design Patterns: Elements of Reusable Object-Oriented Software, design patterns have been used to design programs and ease their maintenance, to teach object-oriented concepts and related “good” practices in classrooms, and to assess quality and help program comprehension in research. However, design patterns may also lead to overengineered programs and may negatively impact quality. We recall the history of design patterns and present some recent development characterizing the advantages and disadvantages of design patterns. Design patterns are a form of documentation that proposes solutions to recurring object-oriented software design problems. Design patterns became popular in software engineering thanks to the book published in 1995 by the Gang of Four (GoF): Erich Gamma, Richard Helm, Ralph Johnson, and John Vlissides. Since the publication of the book Design Patterns: Elements of Reusable Object-Oriented Software, design patterns have been used to design programs and ease their maintenance, to teach object-oriented concepts and related “good” practices in classrooms, and to assess quality and help program comprehension in research. However, design patterns may also lead to overengineered programs and may negatively impact quality. We recall the history of design patterns and present some recent development characterizing the advantages and disadvantages of design patterns.
Article
Recently, in (Ste05), we proved that an adaptive nite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from (SINUM, 38 (2000), 466{488) by Morin, Nochetto, and Siebert, converges with the optimal rate. The number of triangles N in the output partition of such a method is generally larger than the number M of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes. A key ingredient to our proof was a result from (Numer. Math., 97(2004), 219{268) by Binev, Dahmen and DeVore saying that N N0 CM for some absolute constant C, where N0 is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of n-simplices, with that generalizing the result concerning optimality of the adaptive nite element method to general space dimensions.
varfem: variational formulation based programming for finite element methods in matlab
  • Y Yu
Y. Yu, varfem: variational formulation based programming for finite element methods in matlab (2022). 2206.06918
iFEM: an integrated finite element methods package in MATLAB
  • L Chen
L. Chen, iFEM: an integrated finite element methods package in MATLAB, Technical Report, 2009. https://github.com/lyc102/ifem.
The fenics project version 1.5
  • Alns
Efficient and flexible MATLAB implementation of 2d and 3d elastoplastic problems
  • M Ermák
  • S Sysala
  • J Valdman
M. Č ermák, S. Sysala, J. Valdman, Efficient and flexible MATLAB implementation of 2d and 3d elastoplastic problems, Appl. Math. Comput. 355 (2019) 595-614, doi: 10.1016/j.amc.2019.02.054.
The dune framework: Basic concepts and recent developments
  • P Bastian
  • M Blatt
  • A Dedner
  • N.-A Dreier
  • C Engwer
  • R Fritze
  • C Grser
  • C Grninger
  • D Kempf
  • R Klfkorn
  • M Ohlberger
  • O Sander
P. Bastian, M. Blatt, A. Dedner, N.-A. Dreier, C. Engwer, R. Fritze, C. Grser, C. Grninger, D. Kempf, R. Klfkorn, M. Ohlberger, O. Sander, The dune framework: Basic concepts and recent developments, Comput. Math. Appl. 81 (2021) 75-112, doi: 10.1016/j.camwa.2020.06.007.