An Introduction to the Gradient Discretisation Method

2019 ◽  
pp. 451-459
Author(s):  
Jérôme Droniou ◽  
Robert Eymard ◽  
Thierry Gallouët ◽  
Cindy Guichard ◽  
Raphaèle Herbin
Keyword(s):  
Gradient Discretisation Method

scholarly journals The Gradient Discretisation Method for Linear Advection Problems

2020 ◽  
Vol 20 (3) ◽  
pp. 437-458 ◽  
Author(s):  
Jérôme Droniou ◽  
Robert Eymard ◽  
Thierry Gallouët ◽  
Raphaèle Herbin
Keyword(s):  
Convergence Analysis ◽  
Hyperbolic Equations ◽  
Parabolic Partial Differential Equations ◽  
Numerical Tests ◽  
Simultaneous Design ◽  
Conforming Finite Element ◽  
Gradient Discretisation Method ◽  
Volume Method ◽  
Polytopal Meshes ◽  
Linear Scalar

AbstractWe adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method.

scholarly journals The Gradient Discretisation Method

2018 ◽  
Author(s):  
Jérôme Droniou ◽  
Robert Eymard ◽  
Thierry Gallouët ◽  
Cindy Guichard ◽  
Raphaèle Herbin
Keyword(s):  
Gradient Discretisation Method

A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yahya Alnashri ◽  
Hasan Alzubaidi
Keyword(s):  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Finite Element Scheme ◽  
Anisotropic Reaction Diffusion ◽  
Gradient Discretisation Method ◽  
Glioma Tumors ◽  
The Brain ◽  
Anisotropic Case ◽  
Methods Numerical ◽  
Diffusion Problems

Abstract A gradient discretisation method (GDM) is an abstract setting that designs the unified convergence analysis of several numerical methods for partial differential equations and their corresponding models. In this paper, we study the GDM for anisotropic reaction–diffusion problems, based on a general reaction term, with Neumann boundary condition. With natural regularity assumptions on the exact solution, the framework enables us to provide proof of the existence of weak solutions for the problem, and to obtain a uniform-in-time convergence for the discrete solution and a strong convergence for its discrete gradient. It also allows us to apply non-conforming numerical schemes to the model on a generic grid (the non-conforming ℙ ⁢ 1 {\mathbb{P}1} finite element scheme and the hybrid mixed mimetic (HMM) methods). Numerical experiments using the HMM method are performed to assess the accuracy of the proposed scheme and to study the growth of glioma tumors in heterogeneous brain environment. The dynamics of their highly diffusive nature is also measured using the fraction anisotropic measure. The validity of the HMM is examined further using four different mesh types. The results indicate that the dynamics of the brain tumor is still captured by the HMM scheme, even in the event of a highly heterogeneous anisotropic case performed on the mesh with extreme distortions.

scholarly journals Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method

2018 ◽  
Vol 18 (4) ◽  
pp. 609-637 ◽  
Author(s):  
Jérome Droniou ◽  
Neela Nataraj ◽  
Devika Shylaja
Keyword(s):  
Optimal Control ◽  
Finite Elements ◽  
Finite Difference Methods ◽  
Mixed Finite Elements ◽  
Neumann Boundary ◽  
Optimal Order ◽  
Active Set ◽  
Order Error ◽  
Wide Range ◽  
Gradient Discretisation Method

AbstractThe article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.

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