Well posedness for the Poisson problem on closed Lipschitz manifolds

We study the weak formulation of the Poisson problem on closed Lipschitz manifolds. Lipschitz manifolds do not admit tangent spaces everywhere and the definition of the Laplace–Beltrami operator is more technical than on classical differentiable manifolds (see, e.g., Gesztesy in J Math Sci 172:279–346, 2011). They however arise naturally after the triangulation of a smooth surface for computer vision or simulation purposes. We derive Stokes’ and Green’s theorems as well as a Poincaré’s inequality on Lipschitz manifolds. The existence and uniqueness of weak solutions of the Poisson problem are given in this new framework for both the continuous and discrete problems. As an example of application, numerical results are given for the Poisson problem on the boundary of the unit cube.


Introduction
Traditionally in the context of the finite element simulation of PDEs on surfaces [see, e.g., [1][2][3][4], a continuous surface is first triangulated into a piecewise linear surface h that is no longer differentiable but merely a Lipschitz manifold.The solution u of a linear PDE on can then be approximated by the solution u h of the same PDE on h .Using a variational formulation, one then seek for ũh the projection of u h on the finite dimensional space generated by the nodal functions of h .The numerical resolution of the resulting linear system yields the nodal values of ũh .
In [1,2], the well-posedness of the Poisson problem − u = f on a smooth surface immersed in R d is studied, and a finite element approximation is proposed.In the context of immersed surfaces, [2] defines differential operators (tangential gradient and divergence) which are based on Fermi coordinates.Fermi coordinates [see Section 2.3 in 2] are global coordinates on a hypersurface ∈ R d that avoid the use of local charts.Unfortunately, the existence of Fermi coordinates requires to be C 2 , or more precisely to satisfy both an interior and an exterior sphere condition.The issue of polyhedral surfaces is raised in [1] (page 3), which quotes [5] for the definition of Sobolev spaces on C 0,1 manifolds.However, [5] deal with C 2,κ manifolds (see Definition 2.10 and Section 4.2).The assumption of C 2 regularity made on the surface in [1,2] restricts the scope of surfaces for which the methodology yields the well-posedness of the continuous problem.Furthermore, even if is assumed C 2 , after triangulation of , the new surface h is piecewise linear, and no longer C 2 .The study and characterisation of u h the solution of the Poisson problem on h can not therefore be performed using the methodology in [1,2] due to the lack of smoothness of h .
In order to perform calculus on more general surfaces that are not C 2 , especially on triangulated surfaces, we can not rely on the approach of [1,2] using Fermi coordinates on C 2 immersed surfaces.We chose instead to extend the definition of the classical differential operators to Lipschitz manifolds following [6] (Sect.3), [7] (Appendix A) and [8].
Lipschitz manifolds are an intermediate structure between topological manifolds where the tangent space is defined nowhere and C 1 manifolds where the tangent space is defined everywhere.On Lipschitz manifolds the tangent space is defined almost everywhere, which is enough to define spaces of functions that are weakly differentiable.Gesztesy et al. [7] defines Lebesgue L p spaces, as well as Sobolev W 1, p and W −1, p spaces.The main technical difficulty is that Lipschitz manifolds are not differentiable everywhere.For that reason, they make an intensive use of differentiability almost everywhere through the Rademacher's theorem.We use this new setting for the definition and study of the Poisson problem on a Lipschitz manifold, and propose a finite element method for its numerical simulation.
The article is organised as follows.In Sect.2, following [7], we recall the definition of Lipschitz manifolds, and the associated differential and exterior calculi.We go on introducing the Sobolev spaces as well as the Laplace-Beltrami operator on a Lipschitz manifold.We extend [7] by proving Stokes' 1 and Green's 2 theorems as well as a Poincaré's inequality on Lipschitz manifolds.In Sect.3, in the context of closed Lipschitz manifold we study the Poisson problem and give a theorem of existence and uniqueness of solutions.This theorem relies on the weak formulation of the problem that is available thanks to the Green's theorem.Section 4 is devoted to the introduction of the linear finite element discretization and Sect. 5 is concerned with the numerical simulation of the Poisson problem on a Lipschitz manifold: the boundary of the unit cube.

Calculus on Lipschitz manifolds
In standard calculus, the gradient, divergence and Laplace operators are classically defined on the Euclidean space R d .They can however also be defined on a C 1 manifold M in an intrinsic way (no immersion into R d ), using a Riemannian metric [see for example Section 1.2 in 9] or the Hodge operator [see Section 11.2 in 10].

Lipschitz manifolds
We recall that a topological manifold M of dimension d ∈ N * is a topological space such that for every x ∈ M there exists an open set U ⊂ M and a map φ : U → R d such that x ∈ U and φ is an homeomorphism onto its image φ(U ).The couple (U , φ), where U ∈ M is an open set around x and φ : We recall that a map φ : E → F between two metric spaces E and F is a bilipschitz map if it is a Lipschitz homeomorphism onto its image φ(E), and its inverse is also Lipschitz.

Definition 1 (Lipschitz manifold)
A Lipschitz manifold is a topological manifold with an atlas (called a Lipschitz atlas) {U i , φ i } i∈I such that for any i, j ∈ I the transition map Since differentiability will occur only almost everywhere, functional analysis on Lipschitz manifolds requires the definition of negligible sets.

Definition 2 (Negligible sets) Let be a Lipschitz manifold and {U
A property that is true on all but a negligible set of points of is said to hold almost everywhere.
Given a Lipschitz manifold with a Lipschitz atlas {U i , φ i } i∈I , a point x ∈ is said to be a singular point if there exists i, j ∈ I such that x ∈ U i ∩ U j and φ i • φ −1 j is not differentiable at φ j (x).A point x that is not singular is called a regular point.
A consequence of the Rademacher's theorem [see Corollary 11.7 in 11] is that the set of singular points is negligible.Hence for any Lipschitz manifold, one can properly define a tangent space only almost everywhere (see Definition 5 below).Note that if the Lipschitz manifold is embedded in R d , it admits almost everywhere a normal vector.
Unlike smooth manifolds where differentiability can take place everywhere and be of any order, differentiability on Lipschitz manifolds can happen only at regular points and is necessarily of order one.
Lipschitz maps are differentiable almost everywhere thanks to the Rademacher's theorem.

Tangent space
Let x ∈ be a regular point.A path through x is a continuous map γ x :] − , [→ , > 0 with γ x (0) = x and such that there exists i ∈ I , x ∈ U i and φ i • γ x is differentiable at 0. We define the following equivalence relation between paths through x: γ 1x and γ 2x are equivalent if they share the same derivative at x: Following [7] appendix A and [6] Section 3.1, we define the tangent space almost everywhere as the space of equivalence classes γx for the relation (1).

Definition 5 (Tangent space)
Let be a Lipschitz oriented manifold.The tangent space of at x ∈ is the vector space If is a Lipschitz oriented manifold of dimension d ∈ N * , its tangent space at every regular point is a vector space of dimension d.At every regular point x ∈ belonging to a local chart (U , φ = (φ 1 , . . ., φ d )), we can define a basis of the tangent space T x by considering the d curves φ On a Lipschitz manifold of dimension d, if x ∈ is a regular point, T x is a real vector space of dimension d and basis φ1 x , . . .φd x .Following a classical duality argument, the strong gradient of f at x, which is a linear form, can therefore be represented by a d-dimensional real vector denoted

Measurable and Lipschitz differential forms
On any vector space E of dimension d, we introduce for any integer l ∈ {0, 1, . . ., d} the l-th exterior power l E as the vector space of l-linear alternate forms on E.
Because of the lack of differentiability of Lipschitz manifolds, we cannot work with smooth differential forms as done classically.Instead, following [6] (Section 3.2), we introduce measurable and Lipschitz differential forms as follows.
A measurable (resp.Lipschitz) differential form ω of degree l is a mapping defined almost everywhere on such that • ω ∈ l T x for almost every x ∈ • for any local chart (U , φ = (φ 1 , . . ., φ d )) and any multi-index J of length l, there exists a measurable (resp.Lipschitz) function a J such that almost everywhere Due to the lack of regularity, we cannot define the exterior derivative as an endomorphism on differential forms as done classically [see, e.g., Theorem 5.42 in 12].Instead, the derivative of a Lipschitz differentiable form is a measurable differentiable form defined as follows.

Definition 8 (Exterior derivative of a Lipschitz differentiable form)
Let be a Lipschitz manifold of dimension d, and {U i , φ i } i∈I its Lipschitz atlas.Let l ∈ {0, 1, . . ., d}, and ω a Lipschitz differential form of degree l on .The exterior derivative of ω is the measurable differential form dω of degree l + 1 such that on any local chart where a J are the Lipschitz coefficients of ω: ω = |J |=l a J dφ J .
In Definition 8, the differentiability of a Lipschitz form follows from the Rademacher's theorem.
The orientability of a manifold is a necessary condition for the existence of a volume form [see, e.g., Theorem 6.5 in 12], which in turns yields a measure and integration theory on the manifold.The classical definition of orientability involves the positiveness everywhere of the Jacobian of the transition maps φ i • φ −1 j .However, in the case of Lipschitz manifolds, the transition maps are differentiable only almost everywhere.

Definition 9 (Orientation) A Lipschitz manifold with Lipschitz atlas {U
An orientation is therefore given by an atlas such that the coordinate charts (U i , φ i = (φ i1 , . . ., φ id )) define a positively oriented basis d dφ i1 , . . ., d dφ id of the tangent space almost everywhere.On any oriented Lipschitz manifold of dimension d, there exists a Lipschitz differentiable form dV (x) ∈ d T x such that dV (x) = 0 for almost every x ∈ .The proof is a direct adaptation of Theorem 6.5 in [12].Such a form is called a volume form [see, e.g., Definition 6.3 in 12] and takes the form dV Integrating volume forms requires the pullback operator.Following the definition 5.18 in [12] we define the pullback of a volume form by a Lipschitz map.The pullback of a measurable differentiable form ω on by g is the measurable differentiable form g * ω on R d defined by Note that the pullback of a Lipschitz differential form by a Lipschitz map is merely a measurable differential form (not Lipschitz).For instance, the pullback of a volume form see, e.g., Proposition 6.12 in 12], which is not necessarily Lipschitz.On R d , volume forms take the form dV where (θ i ) i∈I is a Lipschitz partition of unity on , subordinate to the cover We are now ready to state Stokes' theorem for Lipschitz forms, which is required for the integration by part (see the proof of Green's theorem 2).
Theorem 1 (Stokes' theorem) Let be a compact oriented Lipschitz manifold of dimension d.Let ω be a Lipschitz form of degree d − 1 on .Then Proof Since is compact, there is a finite set I and a finite number of charts (U i , φ i ) i∈I that covers : = ∪ i∈I U i .Let (α i ) i∈I be a smooth partition of unity subordinate to that cover.The existence of (α i ) i∈I is given by Proposition 6.14 in [12].Since ω = i∈I α i ω, it is sufficient to prove the theorem for α i ω that is, for a Lipschitz differentiable form supported within an open subset U i .We assume therefore in the following that supp(ω) ⊂ U i where Since ω is a Lipschitz differentiable form of degree d − 1, for any j ∈ {1, 2, . . ., d}, there exists a Lipschitz function a j supported in U i such that almost everywhere Since ω is Lipschitz, it admits an exterior derivative which is From the definition of the tangent map (Definition 6) we have The pullback of the exterior derivative is From the definition of form integrals (8): Using the tangent map property ( 13) we obtain Since a j is compactly supported within the open set U i , a j is zero in a neighborhood of ∂U i and therefore on ∂U i we have a j = 0 and ∇a j = 0. Similarly on ∂φ i (U i ), a j • φ −1 i = 0 and ∇a j • φ −1 i = 0, where ∇a j denotes the tangent map of a j .∇a j goes from T x to R and can therefore be identified with a vector denoted ∂a j ∂φ 1 , . . ., ∂a j ∂φ d .We can therefore extend a j • φ −1 i to R d as a Lipschitz function f j with compact support: Since a j is Lipschitz and φ −1 i is Lipschitz by definition of a Lipschitz chart, f j is Lipschitz as composition of Lipschitz functions and then is differentiable by Rademacher's Theorem.Now (18) becomes where ( 22) is a consequence of Fubini's theorem.The result follows from the fact that f j has compact support.

Sobolev spaces on Lipschitz manifolds
The Lebesgue spaces are defined by a pullback of the Euclidean Lebesgue space.
The space L p ( ) is defined as the set of real valued functions f defined almost everywhere on such that for any local chart . The associated norm is The integral above is defined through a volume form (see formula 8).
The strong gradient − → ∇ x φ was defined in (6) for a differentiable function f at a regular point x.The usual definition of Sobolev Spaces [see 13] is based on smooth test functions.Since differentiability is limited to order one on Lipschitz manifolds we choose to replace smooth test functions by Lipschitz test functions.A function f ∈ L p ( ) is said to be weakly differentiable if there exists g ∈ L p ( ) d such that for any Lipschitz function φ g is called the weak gradient of f and denoted − → ∇ f .Because Lipschitz manifolds admit only one order of differentiability, we can study only the space of order one weakly differentiable functions.The Sobolev space W 1, p ( ) is defined in a similar way to the Euclidean case.
Definition 12 (Sobolev space W 1, p ( )) Let p ∈ [1, ∞], be a compact oriented Lipschitz manifold.The space W 1, p ( ) is defined as the set of weakly differentiable functions equipped with the norm We introduce the following classical notations in the Hilbertian case: From the Definition 12 of Sobolev spaces, it is straightforward that The definition of the weak divergence by a duality approach as well as the study of weak solution of the Poisson problem both require Sobolev spaces with negative index.Following [13] sections 3.7 to 3.13, we define W −1, p ( ) using duality.
Definition 13 (Dual Sobolev space W −1, p ( )) Let p ∈ [1, ∞[, be a compact oriented Lipschitz manifold.The space W −1, p ( ) is defined as the dual of the space W 1, p ( ): W −1, p ( ) = (W 1, p ( )) , where p is the conjugate of p: W −1, p ( ) is the extension to W 1, p ( ) of distributions that act normally on infinitely smooth test functions [see Section 3.10 in 13].
The weak divergence is defined for p ∈]1, ∞] as the adjoint of the weak gradient The surface divergence of a general function f ∈ L p ( ) d is therefore a linear operator acting on W 1, p ( ).However, for f ∈ W 1, p ( ) d , ∇ • f can be identified by duality with a function in L p ( ).
The following Green's theorem connects the divergence and the gradient on Lipschitz manifolds.From a functional analytic point of view, it is an extension of the definition of weak differentiability (24) to test functions in W 1, p ( ).

Theorem 2 (Green's theorem) Let be a compact oriented Lipschitz manifold of dimension
Proof Define the following Lipschitz differential forms of order (d − 1): where v i , i = 1, . . ., d are the components of v. Since Given that in a similar way to the strong gradient, the weak gradient satisfies the product rule the result follows from Stokes theorem 1: A consequence of the Green's theorem 2 is that the operator − is symmetric and positive.Green's theorem 2 will be used to perform the integration by part needed to obtain the weak formulation (38) of the Laplace-Beltrami operator (Theorem 3).

The Laplace-Beltrami operator on Lipschitz manifolds
The Laplace-Beltrami operator is classically defined as a second order differential operator on C 2 manifolds [see for instance 10].The lack of smoothness on a Lipschitz manifold prevents us from making sense of the Laplace-Beltrami operator as a differential operator of order two.One way of defining the Laplace-Beltrami operator on a Lipschitz manifold is The following theorem, taken from [7] (Theorem 1.3) gives some important properties of the Laplace-Beltrami operator on Lipschitz manifolds on the Sobolev space Theorem 3 (Laplace-Beltrami operator on Lipschitz manifolds) Let be a compact, connected, oriented Lipschitz manifold.The operator := ∇ • − → ∇ is well defined, self adjoint and bounded from W 1, p ( ) to W −1, p ( ).Moreover the operator − has a purely discrete spectrum with λ j → ∞ as j → ∞.Furthermore, there is an Hilbertian basis (ψ j ) j≥0 of L 2 ( ) composed of eigenvectors A corollary of Theorem 3 is the existence of the Poincaré constant on H 1 ( ) which is the first non zero eigenvalue λ 1 of − .

Corollary 1 (Poincaré's inequality) Let be a compact, connected, oriented Lipschitz manifold. There exists a constant C
(32) A classical consequence of Poincaré's inequality is that provided u = 0, the L 2 norm of ∇ u is equivalent to the H 1 norm of u:

The Poisson problem on a closed Lipschitz surface
Now that we have defined the Laplace-Beltrami operator in Sect.2.5, we can study the Poisson problem on closed Lipschitz manifolds.We start by setting the problem and the relevant functional spaces in Sect.3.1.We then give the weak formulation of the problem in Sect.3.2.We end up by giving an existence result in Sect.3.3.

Definition and functional spaces
Let be a closed Lipschitz manifold.Since is closed, it has no boundary (∂ = ∅), and all the constant functions u are in the kernel of . is therefore not invertible on the space of functions u ∈ H 1 ( ).
We choose to impose the global condition u = 0 to guarantee the uniqueness of solution.Hence we define Consider the following Poisson problem for a given f ∈ L 2 0 ( ): In the next section we give the weak form of the Poisson problem (35).

Weak form of the Poisson problem
The classical Laplace-Beltrami operator acts on C 2 functions.A classical solution of ( 35) is therefore a function u ∈ C 2 ( ).Unfortunately such a strong solution doesn't always exists even if f is assumed continuous (see Section 3.1.2in [14] in the Euclidean case).We now define a weak form of the Laplace Beltrami operator which acts on H 1 0 ( ) instead of C 2 ( ).The weak Laplace-Beltrami operator on : is the operator sending u ∈ H 1 0 ( ) to the linear functional where − → ∇ u is the weak gradient of u on given by ( 24).The variational formulation for (35) is the following.
A solution u ∈ H 1 0 ( ) of ( 38) is called a weak solution of the Poisson problem (35).Indeed, it is only one time weakly differentiable whereas a strong (classical) solution is twice strongly differentiable.
In the next section we are going to prove following [2], that the Poisson problem with a zero mean right hand side admits, a unique solution with zero mean on a closed Lipschitz manifold.

Existence result
The existence and uniqueness of weak solutions for the variational formulation (38) of problem (35) on non smooth manifolds is to our knowledge open.Classical existence results require the smoothness of [see for instance Chapter 4, Section 1.2 in 10].The following statement is taken from [1] Theorem 1 b).
Theorem 4 (Existence and uniqueness of weak solutions on a C 3 hypersurface [1]) Let be a closed embedded C 3 hypersurface in R 3 .For every f ∈ L 2 0 ( ), there exists a unique weak solution u ∈ H 1 0 ( ) of − u = f on .
Unfortunately on a Lipschitz manifold, similar existence results can not be found in the literature.Besides, as mentioned in [1] page 3, it is important to check that the space H 1 ( h ) is well defined, where h results from a triangulation of .
The following statement is taken from [1] Theorem 2 b).
Theorem 5 (Existence and uniqueness of weak solutions on a triangulated hypersurface [1]) Let h be a closed embedded C 0,1 hypersurface in R 3 .For every f h ∈ L 2 ( h ) with h f h = 0, there exists a weak solution u h ∈ H 1 ( h ) of − u h = f h on h .Furthermore u h is unique up to a constant.
Dziuk [1] claims that the proof is a "simple application of usual Hilbert space methods".However these Hilbert space methods require a geometrical and functional setting for calculus on Lipschitz manifolds that is not easily found in the literature.Moreover Stokes' theorem 1 and Poincaré's inequality (Corollary 1) on Lipschitz manifolds are never stated and do not seem trivial to us.Our existence theorem on Lipschitz manifolds relies on the Stokes' theorem 1 and on the Lax-Milgram's theorem.
Theorem 6 (Existence theorem on Lipschitz manifolds) Let be a closed Lipschitz manifold.Let f ∈ L 2 0 ( ).There exists a unique weak solution u ∈ H 1 0 ( ) of − u = f on .Furthermore, u depends continuously on the right hand side: Proof In order to use the Lax-Milgram's theorem, using the bilinear form we rewrite the weak formulation (38) of (35) as • We obtain the coercivity of a(•, •) thanks to the Poincaré's inequality (Corollary 1) or more precisely inequality (33): • The continuity of b(•) is the consequence of the Cauchy-Schwarz's inequality and the Poincaré's inequality (Corollary 1) Since the requirement of the Lax-Milgram's theorem are satisfied, the existence of a unique solution to the weak formulation (38) satisfying the stability estimate (39) holds with C = 1 (1+C) 2 .
In the next section we introduce the linear finite element method.

The finite element method
The finite element method (FEM) is a numerical method for solving problems of engineering and mathematical physics that are formulated by PDEs [see, e.g., 3,14]).
When the domain is a hypersurface , we assume as done in [1], the existence of a polyhedric surface h that approximates and is made of triangles with nodes on .We can approximate functions u ∈ H 1 0 ( ) by functions u h ∈ H 1 0 ( h ).We can then approximate functions u h ∈ H 1 0 ( h ) by their projection on P L 0 ( h ) the space of continuous piecewise affine functions with zero mean.
Considering the weak Laplace-Beltrami operator on the piecewise linear manifold h , we seek for ũh ∈ P L 0 ( h ) the projection on P L 0 ( h ) of the solution u h ∈ H 1 0 ( h ) of the Poisson problem − h u h = f h .The finite element approximation therefore projects the original problem set on H 1 0 ( ) onto the finite dimensional space P L 0 ( h ) using a discrete variational formulation.
The discrete variational formulation of the Poisson equation ( 35) is the following.
Find ũh ∈ P L 0 ( h ) such that ∀ ṽh ∈ P L 0 ( h ), In a similar fashion as in Eqs.(36, 37), the discrete Laplace-Beltrami operator − ˜ h : P L 0 ( h ) → P L 0 ( h ) is the operator sending ũh ∈ P L 0 ( h ) to the linear functional where − → ∇ h ũh is the weak gradient of ũh on h given by (24).− ˜ h acts between finite dimensional spaces and approximates the continuous weak Laplace-Beltrami operator (36, 37).It can be represented by a stiffness matrix A h with size n the number of nodes of h .
The unknown function ũh belongs to the finite dimensional space P L 0 ( h ).Define the unknown vector (u 1 , ..., u n ) as the vector of components of ũh on the nodal basis: Since P L 0 ( h ) is generated by the nodal functions φ i : h → R, i = 1, ..., n such that φ i (x j ) = δ i j , the solution ũh of the discrete Poisson problem (40) must therefore satisfy the following system of equations ∀i ∈ {1, ..., n}, which takes the algebraic form From (44) the coefficients of the stiffness matrix A h = (a i j ) i, j=1,...,n , and of the right hand side vector b h = t (b 1 , ..., b n ) are therefore given by The stiffness matrix A h is symmetric positive but not invertible [see Theorem 4.1 in 4]).However the finite element linear system admits a unique solution provided the right hand side has zero mean [see Theorem 4.2 in 4], hence the existence of the discrete solution ũh as stated in the following theorem.
Theorem 7 (Existence of the discrete solution) Let f h ∈ P L 0 ( h ), there exists a unique discrete solution ũh ∈ P L 0 ( h ) to the discrete Poisson problem (40).
The computation of the coefficients a i j is usually performed by expressing the integral in (46) as a sum over the triangles T k composing h : The gradients − → ∇ h φ i in (48) are thus computed in each separate triangle T k where their value is well defined and constant.There is no need to compute the gradient at the edge between two triangles T k , where it is not properly defined.Therefore the lack of smoothness of h does not yield any singularity in the computation of the stiffness matrix coefficients.
The coefficients a i j of A h can be computed explicitly as a function of the mesh nodes coordinates and are given in [4].

Numerical experiments on the unit cube boundary
In order to illustrate the applicability of our theoretical study we present in this section the numerical simulation of a Poisson problem on the boundary of the unit cube.
Fig. 1 The unit cube in SALOME CAO module is a topological manifold but not a differential manifold because of the presence of sharp edges where admits no tangent space. is however a Lipschitz manifold,which admits tangent spaces except at the edges.As a Lipschitz manifold, can be endowed with a Laplace-Beltrami operator = ∇ • − → ∇ , defined as the combination of a surface divergence ∇ •, and of a surface gradient − → ∇ [see 7] (Fig. 1).We consider f the restriction to of the smooth function cos(2π x) cos(2π y) cos(2π z).The explicit expressions of f on each face of are f is an eigenfunction of the Laplace-beltrami operator on since We are going to solve the following Poisson problem on : where the right hand side f defined in (49) and the unknown u ∈ H 1 ( ) are zero mean functions.
Our objective is to solve numerically the Poissson problem (35) using the finite element method described in [1].

Meshing of the domain
Below are the meshes used in our convergence analysis (Fig. 2).

Visualisation of the results
Below are visualisations of the numerical results obtained on the different meshes Below are clipings of the previous numerical results (Figs. 3, 4).

Numerical convergence
When refining the mesh, we observe the convergence of the method with a numerical order of 1.91 (Fig. 5).

Conclusion and perspectives
The notion of Lipschitz manifold has been studied by some authors [6,7] and is usefull in laying the foundation of the finite element method on general compact manifolds.We showed how it allows the proper definition of tangent spaces, gradient and divergence operators almost everywhere.Differentiability almost everywhere thanks to Rademacher's theorem is the key ingredient of the calculus on Lipschtz manifolds.We have proposed proofs of the Stokes' (1) and Green's (2) theorems as well as a Poincaré's inequality.This enables the weak formulation of the Poisson problem.The well-posedness of the weak formulation of the Poisson problem is then obtained thanks to the Lax-Milgram's theorem.
We gave an example of numerical simulation on a Lipschitz (non smooth) manifold to illustrate the fact that the lack of smoothness does not yield any numerical singularity.The numerical results of the Poisson problem on the unit cube boundary showed that the finite element approximation converges towards the exact solution.
This study has set the theoretical ground for further numerical analysis of more complex PDEs on Lipschitz manifolds.Note however that on Lipschitz manifolds the maximum order of differentiability is one which is enough for the classical weak formulation of second order elliptic PDEs.This is however a technical difficulty for general high order PDEs on Lipschitz manifolds.One workaround for PDEs of order three and more could however be to reset the PDE into a system of low order PDEs.
Yet, with this new approach we were not able to extend the regularity and convergence theorems of [1,2] because the lack of smoothness of the Lipschitz manifold prohibits the use of the lift operator.Further research should therefore be devoted to the theoretical analysis of the convergence of both the approximate manifold h and solution u h towards the exact manifold and solution u.

Definition 10 (
Pullback of a volume form) Let be a Lipschitz manifold of dimension d ∈ N * , U an open subset of R d and let g : U → be a Lipschitz map.

Fig. 2 Fig. 3
Fig. 2 Meshes of the unit cube boundary

Fig. 4 Fig. 5
Fig. 4 Clipings of the numerical results on the unit cube boundary On a general Lipschitz manifold, the integral of a volume form [see, e.g., Definition 6.16 in 12], is defined by

•
For the numerical resolution of our discrete problem, we use an iterative solver (Conjugate Gradient) because the stiffness matrix A h is large and sparse [see 15].• For the design and meshing of the domain we use the GEOMETRY and MESH modules of the software SALOME 9.5 [see 16, 17].• For the visualisation of the result, we use the PARAVIS module included in SALOME [see 17].• For the coding of the script, we use Python language and the open-source Linux based library [18] which is very practical for the manipulation of large matrices, vectors, meshes and fields.It (SOLVERLAB) can handle finite element and finite volume discretizations, read general 1D, 2D and 3D geometries and meshes generated by SALOME.