Gradient discretisation method: Difference between revisions

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{{Short description|Method for numerical differential equations}}

<!-- {{more footnotes|date=March 2017}} answer and improvement completed by Cyclotourist -->

[[Image:Plaplacien4.svg|thumb|right|400px|Exact solution <br/> <math>\overline{u}(x) = \frac 3 4 \left({0.5}^{4/3}- |x - 0.5|^{4/3}\right)</math> <br/>

of the ''p''-Laplace problem <math>-( |\overline{u}'|^2 \overline{u}')'(x) = 1</math> on the ___domain [0,1] with <math>\overline{u}(0) = \overline{u}(1) = 0</math>

(black line) and approximate one (blue line) computed with the first degree discontinuous Galerkin method plugged into the GDM (uniform mesh with 6 elements).]]

{{Differential equations}}

In [[numerical mathematics]], the '''gradient discretisation method''' ('''GDM)''') is a framework which contains classical and recent discretisationnumerical schemes for diffusion problems of differentvarious kinds: linear or non -linear, steady-state or time-dependent. The schemes may be conforming or non -conforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless).

Some core properties are required to prove the convergence of a GDM. These core properties enable complete proofs of convergence of the GDM for elliptic and parabolic problems, linear or non-linear. For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM <ref>'''R. Eymard, C. Guichard, and [[Raphaèle Herbin|R. Herbin]].''' Small-stencil 3d schemes for diffusive flows in porous media. M2AN, 46:265–290, 2012.</ref> (the quantities <math>C_{D}</math>, <math>S_{D}</math> and <math>W_{D}</math>, [[#The example of a linear diffusion problem|see below]]). For non-linear problems, the proofs are based on compactness techniques and do not require any non-physical strong regularity assumption on the solution or the model data.<ref name=droniou>'''J. Droniou, R. Eymard, T. Gallouët, and [[Raphaèle Herbin|R. Herbin]].''' Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. (M3AS), 23(13):2395–2432, 2013.</ref> [[#Some non-linear problems with complete convergence proofs of the GDM|Non-linear models]] for which such convergence proof of the GDM have been carried out comprise: the [[Stefan problem]] which is modelling a melting material, two-phase flows in porous media, the [[Richards equation]] of underground water flow, the fully non-linear Leray—Lions equations.<ref>'''J. Leray and J. Lions.''' Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France, 93:97–107, 1965.</ref>

Some core properties are required to prove the convergence of a GDM. Owing to these core properties, it is possible to prove the convergence of a GDM for standard elliptic and parabolic problems, linear or non-linear. Then any scheme entering the GDM framework is known to converge on these problems; this occurs in the case of the conforming Finite Elements, the Raviart—Thomas Mixed Finite Elements, or the <math>P^1</math> non-conforming Finite Elements, or in the case of more recent schemes, such as the Hybrid Mixed Mimetic or Nodal Mimetic methods, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes.

Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to [[#Galerkin methods and conforming finite element methods|conforming Finite Elements]], [[#Mixed finite element|Mixed Finite Elements]], [[#Nonconforming finite element|nonconforming Finite Elements]], and, in the case of more recent schemes, the [[#Discontinuous Galerkin method|Discontinuous Galerkin method]], [[#Mimetic finite difference method and nodal mimetic finite difference method|Hybrid Mixed Mimetic method, the Nodal Mimetic Finite Difference method]], some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes

==The example of a linear diffusion problem==

Let us considerConsider [[Poisson's equation]] in a bounded open ___domain <math>\Omega\subset \mathbb{R}^d</math>, with homogeneous [[Dirichlet boundary condition]]

{{NumBlk|:|<math>\quad (1) \qquad \qquad -\Delta \overline{u} = f,</math>|{{EquationRef|1}}}}

where <math>f\in L^2(\Omega)</math>,. andThe theusual sense of weak solution <mathref>\overline{u}\in '''H^1_0(\Omega). Brezis.''' Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.</mathref> isto suchthis thatmodel is:

{{NumBlk|:|<math>\quadmbox{Find }\overline{u}\in H^1_0(2\Omega) \qquadmbox{ such \qquadthat, \forallfor all } \overline{v} \in H^1_0(\Omega),\qquadquad \int_{\Omega} \nabla \overline{u}(x)\cdot\nabla \overline{v}(x) dx = \int_{\Omega} f(x)\overline{v}(x) dx. </math>|{{EquationRef|2}}}}

In a nutshell, the GDM for such a model consists in selecting a finite-dimensional space and two reconstruction operators (one for the functions, one for the gradients) and to substitute these discrete elements in lieu of the continuous elements in (2). More precisely, the GDM starts by defining a Gradient Discretization (GD), which is a triplet <math>D = (X_{D,0},\Pi_D,\nabla_D)</math>, where:

The GDM is defined by <math>D = (X_{D,0},\Pi_D,\nabla_D)</math>, where:

* the set of discrete unknowns <math>X_{D,0}</math> is a finite dimensional real vector space,

* the function reconstruction <math>\Pi_D~:~X_{D,0}\to L^2(\Omega)</math> is a linear mapping that reconstructs, from an element of <math>X_{D,0}</math>, a function over <math>\Omega</math>,

* the gradient reconstruction <math>\nabla_D~:~X_{D,0}\to L^2(\Omega)^d</math> is a linear mapping which reconstructs, from an element of <math>X_{D,0}</math>, a "gradient" (vector-valued function) over <math>\Omega</math>. This gradient reconstruction must be chosen such that <math>\Vert \cdot \Vert_{D} := \Vert \nabla_D \cdot \Vert_{L^2(\Omega)^d}</math> is a norm on <math>X_{D,0}</math>.

The related Gradient Scheme for the approximation of (2) is given by: find <math>u\in X_{D,0}</math> such that

{{NumBlk|:|<math>\quad (3) \qquad \qquad \forall v \in X_{D,0},\qquad \int_{\Omega} \nabla_D u(x)\cdot\nabla_D v(x) dx = \int_{\Omega} f(x)\Pi_D v(x) dx. </math>|{{EquationRef|3}}}}

The GDM is then in this case a nonconforming method for the approximation of (2), which includes the nonconforming finite element method. Note that the converse is not true, in the sense that the GDM framework includes methods such that the function <math>\nabla_D u</math> cannot be computed from the function <math>\Pi_D u</math>.

Then there holds the following error estimate, inspired by [Strang,1972]

*The following error estimate, inspired by G. Strang's second lemma,<ref>'''Strang, G. Strang.''' (1972) "Variational crimes in the finite element method".'' In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., pUniv.&nbsp; Maryland, Baltimore, Md., 1972)'', pages 689–710. Academic Press, New York, 1972.</ref> holds

:<math>\quad (4) \qquad \qquad \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d}

\le W_D(\nabla \overline{u}) + 2 S_D(\overline{u}), </math>

{{NumBlk|:|<math>\quad W_D(4)\nabla \qquad overline{u}) \qquad le \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d}

\le W_D(\nabla \overline{u}) + 2 S_D(\overline{u}), </math>|{{EquationRef|4}}}}

and

{{NumBlk|:|<math>\quad (5) \qquad \qquad \Vert \overline{u} - \Pi_D u_D\Vert_{L^2(\Omega)}

\le C_D W_D(\nabla \overline{u}) + (C_D+1)S_D(\overline{u}), </math>|{{EquationRef|5}}}}

defining:

{{NumBlk|:|<math>\quad (6) \qquad \qquad C_D = \max_{v\in X_{D,0}\setminus\{0\}}\frac {\Vert \Pi_D v\Vert_{L^2(\Omega)}} {\Vert \nabla_D v \Vert_{DL^2(\Omega)^d}}, </math>|{{EquationRef|6}}}}

which measures the coercivity (discrete Poincaré constant),

{{NumBlk|:|<math>\forall \varphi\in H^1_0(\Omega),\,

:<math>\quad (7) \qquad \qquad

S_{D}(\varphi) = \min_{v\in X_{D,0}}\left(\Vert\Pi_D v - \varphi\Vert_{L^2(\Omega)} + \Vert\nabla_D v -\nabla\varphi\Vert_{L^2(\Omega)^d}\right), </math>|{{EquationRef|7}}}}

\forall \varphi\in H^1_0(\Omega),\,

which measures the interpolation error,

S_{D}(\varphi) = \min_{v\in X_{D,0}}\left(\Vert\Pi_D v - \varphi\Vert_{L^2(\Omega)} + \Vert\nabla_D v -\nabla\varphi\Vert_{L^2(\Omega)^d}\right), </math>

{{NumBlk|:|<math>\forall \varphi\in H_{\rm operatorname{div}(\Omega),\,

:<math>\quad (8) \qquad \qquad

W_{D}W_D(\varphi) = \sup_max_{uv\in X_{D,0}\setminus\{0\}}\frac{

\forall \varphi\in H_{\rm div}(\Omega),\,

\left|\int_\Omega \left(\nabla_D uv(x)\cdot\varphi(x) + \Pi_D uv(x) \operatorname{\rm div}\varphi(x)\right) \, dx \right|}{\Vert \nabla_D uv \Vert_{DL^2(\Omega)^d}}, </math>|{{EquationRef|8}}}}

W_{D}(\varphi) = \sup_{u\in X_{D,0}\setminus\{0\}}\frac{

which measures the defect of conformity.

\left|\int_\Omega \left(\nabla_D u(x)\cdot\varphi(x) + \Pi_D u(x) {\rm div}\varphi(x)\right) dx \right|}{\Vert u \Vert_{D}} </math>

Note that the following upper and lower bounds of the approximation error can be derived:

Then the core properties which are sufficient for the convergence of the method are, for a family of GDM, that <math>C_D</math> remains bounded, that, for all <math>\varphi\in H^1_0(\Omega)</math>, <math>S_{D}(\varphi)</math> tends to 0, and that for all <math>\varphi\in H_{\rm div}(\Omega)</math>, <math>W_{D}(\varphi)</math> tends to 0. These core properties are not sufficient for proving the convergence of the GDM when applied to some nonlinear problems (nonlinear diffusion, degenerate parabolic problems...),

and we detail in the next section the set of core properties which are required for covering a larger range of problems.

{{NumBlk|:|<math>\begin{align}

&&\frac 1 2 [S_D(\overline{u}) + W_D(\nabla \overline{u})] \\

&\le & \Vert \overline{u} - \Pi_D u_D\Vert_{L^2(\Omega)} + \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d} \\

&\le &(C_D+2) [S_D(\overline{u}) + W_D(\nabla \overline{u})].

\end{align} </math>|{{EquationRef|9}}}}

Then the core properties which are necessary and sufficient for the convergence of the method are, for a family of GDs, the coercivity, the GD-consistency and the limit-conformity properties, as defined in the next section. More generally, these three core properties are sufficient to prove the convergence of the GDM for linear problems and for some nonlinear problems like the <math>p</math>-Laplace problem. For nonlinear problems such as nonlinear diffusion, degenerate parabolic problems..., we add in the next section two other core properties which may be required.

==The core properties allowing for the convergence of a GDM==

These properties are defined for a familyLet <math>(D_m)_{m\in\mathbb{N}}</math> be a family of GDMGDs, defined as above (generally associated with a sequence of regular meshes whose size tends to 0).

=== Coercivity ===

The sequence <math>(C_{D_m})_{m\in\mathbb{N}}</math> (defined by ({{EquationNote|6}})) remains bounded.

=== GD-consistency ===

For all <math>\varphi\in H^1_0(\Omega)</math>, <math>\lim_{m\to\infty} S_{D_m} (\varphi) = 0</math> (defined by ({{EquationNote|7}})).

=== Limit-conformity ===

For all <math>\varphi\in H_\operatorname{\rm div}(\Omega)</math>, <math>\lim_{m\to\infty} W_{D_m}(\varphi) = 0</math> (defined by ({{EquationNote|8}})).

This property implies the coercivity property.

=== Compactness (needed for some nonlinear problems)===

For all sequence <math>(u_m)_{m\in\mathbb{N}}</math> such that <math>u_m \in X_{D_m,0} </math> for all <math>m\in\mathbb{N}</math> and <math>(\Vert u_m \Vert_{D_m})_{m\in\mathbb{N}}</math> is bounded, then the sequence <math>(\Pi_{D_m} u_m)_{m\in\mathbb{N}}</math> is relatively compact in <math>L^2(\Omega)</math> (this property implies the coercivity property).

=== Piecewise constant reconstruction (needed for some nonlinear problems)===

Let <math>D = (X_{D,0}, \Pi_D,\nabla_D)</math> be a gradient discretisation as defined above.

The operator <math>\Pi_D</math> is a piecewise constant reconstruction if there exists a basis <math>(e_i)_{i\in B}</math> of <math>X_{D,0}</math> and a family of disjoint subsets <math>(\Omega_i)_{i\in B}</math> of <math>\Omega</math> such that <math display="inline">\Pi_D u = \sum_{i\in B}u_i\chi_{\Omega_i}</math> for all <math display="inline">u=\sum_{i\in B} u_i e_i\in X_{D,0}</math>, where <math>\chi_{\Omega_i}</math> is the characteristic function of <math>\Omega_i</math>.

==ReviewSome of somenon-linear problems whichwith maycomplete beconvergence approximatedproofs byof athe GDM==

We pass into review a fewsome problems for which the GDM can be proved to converge when the above core properties are satisfied.

=== Nonlinear stationnarystationary diffusion problems ===

:<math>-\quad \qquad \qquad -operatorname{\rm div}(\Lambda(\overline{u})\nabla \overline{u}) = f</math>

In this case, the GDM converges under the coercivity, GD-consistency, limit-conformity and compactness properties.

=== <math>''p</math>''-Laplace problem for ''p'' <math>p> 1</math>===

:<math>-\quad \qquad \qquad -operatorname{\rm div}\left(|\nabla \overline{u}|^{p-2}\nabla \overline{u}\right) = f</math>

In this case, the core properties must be written, replacing <math>L^2(\Omega)</math> by <math>L^p(\Omega)</math>, <math>H^1_0(\Omega)</math> by <math>W^{1,p}_0(\Omega)</math> and <math>H_\operatorname{\rm div}(\Omega)</math> by <math>W_\operatorname{\rm div}^{p'}(\Omega)</math> with <math display="inline">\frac 1 {p} +\frac 1 {p'}=1</math>, and the GDM converges only under the coercivity, GD-consistency and limit-conformity properties.

=== Linear and nonlinear heat equation ===

:<math>\quad \qquad \qquad \partial_t \overline{u}- \operatorname{\rm div}(\Lambda (\overline{u}) \nabla \overline{u}) = f</math>

In this case, the GDM converges under the coercivity, GD-consistency (adapted to space-time problems), limit-conformity and compactness (for the nonlinear case) properties.

=== Degenerate parabolic problems ===

Line 95 ⟶ 114:

Assume that <math>\beta</math> and <math>\zeta</math> are nondecreasing Lipschitz continuous functions:

:<math>\quad \qquad \qquad \partial_t \beta(\overline{u})-\Delta \zeta(\overline{u}) = f</math>

Note that, for this problem, the piecewise constant reconstruction property is needed, in addition to the coercivity, GD-consistency (adapted to space-time problems), limit-conformity and compactness properties.

==Review of some numerical methods which are GDM==

All the methods below satisfy the first four core properties of GDM (coercivity, GD-consistency, limit-conformity, compactness), and in some cases the fifth one (piecewise constant reconstruction).

===[[Galerkin methodsmethod]]s and conforming Finitefinite Elementelement methods===

Let <math>V_h\subset H^1_0(\Omega)</math> be spanned by the finite basis <math>(\psi_i)_{i\in I}</math>. The [[Galerkin method]] in <math>V_h</math> is identical to the GDM where one defines

*<math>X_{D,0} = \{ u = (u_i)_{i\in I} \} = \mathbb{R}^I,</math>,

*<math>\Pi_D u = \sum_{i\in I} u_i \psi_i</math>

*<math>\nabla_D u = \sum_{i\in I} u_i \nabla\psi_i.</math>.

In this case, <math>C_D</math> is the constant involved in the continuous Poincaré's inequality, and, for all <math>\varphi\in H_{\rm operatorname{div}(\Omega)</math>, <math>W_{D}(\varphi) = 0</math> (defined by ({{EquationNote|8}})). Then ({{EquationNote|4}}) and ({{EquationNote|5}}) are implied by [[Céa's lemma]].

The "mass-lumped" <math>P^1</math> finite element case enters in the framework of the GDM:, it suffices to replacereplacing <math>\Pi_D u</math> by <math display="inline">\widetilde{\Pi}_D u = \sum_{i\in I} u_i \chi_{\Omega_i}</math>, where <math>\Omega_i</math> is a dual cell centred on the vertex indexed by <math>i\in I</math>. Using mass lumping allows to get the piecewise constant reconstruction property.

=== Nonconforming <math>P^1</math> finite element ===

On a mesh <math>T</math> which is a conforming set of simplices of <math>\mathbb{R}^d</math>, the nonconforming <math>P^1</math> finite elements are defined by the basis <math>(\psi_i)_{i\in I}</math> of the functions which are affine in any <math>K\in T</math>, and whose value at the centre of gravity of one given face of the mesh is 1 and 0 at all the others (these finite elements are used in [Crouzeix ''et al'']<ref>'''M. Crouzeix and P.-A. Raviart.''' Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7(R-3):33–75, 1973.</ref> for the approximation of the Stokes and [[Navier-Stokes equations]]). Then the method enters into the GDM framework with the same definition as in the case of the Galerkin method, except for the fact that <math>\nabla\psi_i</math> must be understood as the "broken gradient" of <math>\psi_i</math>, in the sense that it is the piecewise constant function equal in each simplex to the gradient of the affine function in the simplex.

In this case, <math>C_D</math> is the constant involved in the continuous Poincaré's inequality, and, for all <math>\varphi\in H_{\rm div}(\Omega)</math>, <math>W_{D}(\varphi) = 0</math> (defined by (8)).

=== Mixed Finitefinite Elementelement ===

The "mass-lumped" <math>P^1</math> finite element case enters in the framework of the GDM: it suffices to replace <math>\Pi_D u</math> by <math>\widetilde{\Pi}_D u = \sum_{i\in I} u_i \chi_{\Omega_i}</math>, where <math>\Omega_i</math> is a dual cell centred on the vertex indexed by <math>i\in I</math>. Using mass lumping allows to get the piecewise constant reconstruction property.

The [[mixed finite element method]] consists in defining two discrete spaces, one for the approximation of <math>\nabla \overline{u}</math> and another one for <math>\overline{u}</math>.<ref>'''P.-A. Raviart and J. M. Thomas.''' A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods ''(Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975)'', pages 292–315. Lecture Notes in Math., Vol. 606. Springer, Berlin, 1977.</ref>

=== Nonconforming <math>P^1</math> finite element ===

It suffices to use the discrete relations between these approximations forto definingdefine a GDM. Using the low degree Raviart-Thomas[[Raviart–Thomas mixedbasis finite elementsfunctions]] allows to get the piecewise constant reconstruction property.

=== Discontinuous Galerkin method ===

On a mesh <math>T</math> which is a conforming set of simplices of <math>\mathbb{R}^d</math>, the nonconforming <math>P^1</math> finite elements are defined by the basis <math>(\psi_i)_{i\in I}</math> of the functions which are affine in any <math>K\in T</math>, and whose value at the centre of gravity of one given face of the mesh is 1 and 0 at all the others. Then the method enters into the GDM framework with the same definition as in the case of the Galerkin method, except the fact that <math>\nabla\psi_i</math> must be understood as the "broken gradient" of <math>\psi_i</math>, in the sense that it is the piecewise constant function equal in each simplex to the gradient of the affine function in the simplex.

The Discontinuous Galerkin method consists in approximating problems by a piecewise polynomial function, without requirements on the jumps from an element to the other.<ref>'''D. A. Di Pietro and A. Ern.''' Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg, 2012.</ref> It is plugged in the GDM framework by including in the discrete gradient a jump term, acting as the regularization of the gradient in the distribution sense.

=== Mixed Finite Element ===

=== Mimetic Finitefinite Differencedifference method and nodal mimetic Mimeticfinite Finite Differencedifference method ===

The Mixed Finite Element method consists in defining two discrete spaces, one for the approximation of <math>\nabla \overline{u}</math> and another one for <math>\overline{u}</math>.

It suffices to use the discrete relations between these approximations for defining a GDM. Using the low degree Raviart-Thomas mixed finite elements allows to get the piecewise constant reconstruction property.

This family of methods is introduced by [Brezzi ''et al'']<ref>'''F. Brezzi, K. Lipnikov, and M. Shashkov.''' Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal., 43(5):1872–1896, 2005.</ref> and completed in [Lipnikov ''et al''].<ref>'''K. Lipnikov, G. Manzini, and M. Shashkov.''' Mimetic finite difference method. J. Comput. Phys., 257-Part B:1163–1227, 2014.</ref> It allows the approximation of elliptic problems using a large class of polyhedral meshes. The proof that it enters the GDM framework is done in [Droniou ''et al''].<ref name=droniou />

=== Mimetic Finite Difference method and nodal Mimetic Finite Difference method ===

==See also==

Line 131 ⟶ 156:

==References==

{{Reflist}}

*'''Strang, G..''' (1972) "Variational crimes in the finite element method" The mathematical foundations of the finite element method with applications to partial differential equations, p.&nbsp;689–710.

== External links ==

* [https://hal.archives-ouvertes.fr/hal-01382358v201382358v7/document The Gradient Discretisation Method] by Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard and [[Raphaèle Herbin]]

{{Numerical PDE|state=expanded}}