Gradient discretisation method: Difference between revisions

Revision as of 06:28, 28 April 2020

Revision as of 20:53, 23 April 2021

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<!-- {{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 \bigleft({0.5}^{4/3}- |x - 0.5|^{4/3}\bigright)</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).]]

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Consider [[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>. The usual sense of weak solution <ref>'''H. Brezis.''' Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.</ref> to this model is:

{{NumBlk|:|<math>\quad (2) \qquad\mbox{Find }\overline{u}\in H^1_0(\Omega)\mbox{ such that, for all } \overline{v} \in H^1_0(\Omega),\quad \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:

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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 reciprocal 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>.

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

{{NumBlk|:|<math>\quad (4) \qquad \qquad W_D(\nabla \overline{u}) \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_{L^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),\,

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>

which measures the interpolation error,

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

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

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

W_D(\varphi) = \max_{v\in X_{D,0}\setminus\{0\}}\frac{

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

which measures the defect of conformity.

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

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

:<math>\quad (9) \qquad \qquad \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>

&&\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.

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

This property implies the coercivity property.

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=== 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>.

==Some non-linear problems with complete convergence proofs of the GDM==

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=== Nonlinear stationary diffusion problems ===

:<math>\quad \qquad \qquad -\operatorname{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.

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=== ''p''-Laplace problem for ''p'' > 1===

:<math>\quad \qquad \qquad -\operatorname{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{div}(\Omega)</math> by <math>W_\operatorname{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{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.

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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.

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===[[Galerkin method]]s and conforming finite element 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>

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*<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 inequality, and, for all <math>\varphi\in H_\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 the framework of the GDM, replacing <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 finite element ===