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A probabilistic approach for exact solutions of determinist PDE's as well as their finite element approximations

02/05/2020
by   Joël Chaskalovic, et al.
0

A probabilistic approach is developed for the exact solution u to a determinist partial differential equation as well as for its associated approximation u^(k)_h performed by P_k Lagrange finite element. Two limitations motivated our approach: on the one hand, the inability to determine the exact solution u to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation u^(k)_h. We thus fill this knowledge gap by considering the exact solution u together with its corresponding approximation u^(k)_h as random variables. By way of consequence, any function where u and u_h^(k) are involved as well. In this paper, we focus our analysis to a variational formulation defined on W^m,p Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation u^(k)_h to consider their respective W^m,p-norm as a random variable, as well as the W^m,p approximation error with regards to P_k finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements P_k_1 and P_k_2, (k_1 < k_2).

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1 Introduction

We recently proposed new perspectives on relative finite element accuracy ([4] and [5]), using a mixed geometrical-probabilistic interpretation of the error estimate in the case of finite element approximation (see for example [7] or [10]), derived from Bramble-Hilbert lemma [1].
In [6], we further extended the results we had derived in the case of to Sobolev spaces . To this end, we had to consider a more general framework, which mainly relied on the Banach-Neas-Babuska (BNB) abstract problem [9] devoted to Banach spaces.
This enabled us to obtain two new probability distributions which estimate the relative accuracy between two Lagrange finite elements and , (), by considering it as a random variable.
We thus obtained new results which show, amongst others, which of or is the most likely accurate, depending on the value of the mesh size ; this value is not considered anymore as going to zero, as in the standard procedure.

However, while obtaining these probability distributions, we only considered the standard error estimate dedicated to

Lagrange finite elements, which approximates the solution to a variational problem (BNB), being formulated in the present case in the Sobolev space .
In the current work, we enrich the model published in [6] in multiple manners. Indeed, considering the functional framework (BNB) in the case of the Sobolev spaces, we will take into account the available a priori estimates one can deal with the solution of this kind of problem as well as for its approximation, together with the approximation error which corresponds to Lagrange finite elements (see Section 2.2). This will enable us to derive a new probabilistic model applied to the relative accuracy between two finite elements and . To this end, we also generalize the discrete probabilistic framework we considered in [4] and [6] by introducing a continuous probabilistic formalism based on an appropriate density of probability (see Section 3.1).
The paper is organized as follows. In Section 2, we recall the mathematical problem we consider and introduce the basic definitions of functional tools to consider different estimations in Sobolev spaces. Section 3.1 is dedicated to the analysis of the relative finite elements accuracy based on a probabilistic approach. In Section 4, we detail the contribution of the a priori estimates and their interactions with the error estimate in the probability distributions we derived in Section 3.1. Concluding remarks follow.

2 The functional and approximation frameworks and their corresponding estimates

2.1 Abstract problem in Banach spaces and corresponding fundamental results

In this section we define an abstract framework which will enable us to consider the solution of a variational problem in Banach Sobolev spaces when , and its corresponding approximation computed by Lagrange finite elements.
In order to do so, we follow the presentation of A. Ern and J. L. Guermond [9], where two general Banach spaces and are involved with reflexive. We also recall the different assumptions needed in order to apply the (BNB) Theorem valid in Banach spaces.
Let be the solution of the variational formulation (VP) defined by:

(1)

where:

  1. and are two Banach spaces equipped with norms denoted by and , respectively; moreover, is reflexive.

  2. is a continuous bilinear form on , i.e,

    with: .

  3. is a continuous linear form on , i.e, .

We further make the two following assumptions:

(BNB1)

,

(BNB2)

.

Then, one can prove the (BNB) Theorem ([9], Theorem 2.6) which claims that variational problem (VP) has one and only one solution in and that the following a priori estimate holds:

(2)

We also define the approximation of , solution to the approximate variational formulation:

(3)

where we assume that and are two finite-dimensional subsets of and , respectively.
Moreover, as noticed in [9] (Remark 2.23, p.92), neither condition (BNB1) nor condition (BNB2) imply its discrete counterpart. Then, the well-posedness of (3) is equivalent to the two following discrete conditions:

(BNB1)

,

(BNB2)

.

If we furthermore assume that dim = dim, a direct application of Theorem 2.2 in [9] enables us to write the following a priori estimate:

(4)

In the next section we will apply these results to the particular case where the exact solution belongs to Sobolev spaces and the approximation is computed by the help of Lagrange finite elements.

2.2 Application to Sobolev spaces and the corresponding error estimate

We introduce an open-bounded subset exactly recovered by a mesh composed by -simplexes which respect the classical rules of regular discretization (see for example [10]). We moreover denote by the mesh size of (the largest diameter in the mesh ), and by the space of polynomials defined on a given -simplex of degree less than or equal to ( 1).
Thenceforth, we assume that the approximate spaces and satisfy dim = dim, that they are included in the space of functions defined on , and composed of polynomials belonging to .
Finally, we also specify the functional framework of the abstract problem (VP) by introducing Sobolev spaces as follows:
For any integer and any , we denote by the Sobolev space of (class of) real-valued functions which, together with all their partial distributional derivatives of order less or equal to , belongs to :

(5)

being a multi-index whose length is given by , and being the partial derivative of order defined by:

(6)

We also consider the norm and the semi-norms , which are respectively defined by:

(7)

where denotes the standard norm in .
Then, in order to fulfill the conditions of the (BNB) Theorem, particularly so that be a Banach space and a reflexive one, in the sequel of the paper, the following definitions of spaces and hold:

(8)

where and are two non zero integers and and two real positive numbers which satisfy and such that:

(9)

Regarding these choices, Sobolev’s space is a Banach space and is a reflexive one [2]. Moreover, we have: and .
We can now recall the a priori error estimate for Lagrange finite elements we derived in [6]:

(10)

where is a positive constant independent of .

Remark 1

Since we noticed that is included in , by considering for the topology of that induced from , thanks to the triangle inequality, (2)- (4) and (10) lead to the following error estimate:

(11)

where here is the dual of .
As one can see, the right-hand side of (11) contains , whose dependency on is usually unknown, except for particular cases. This is the reason why we will assume it is bounded form below by a positive constant independent of h:

This uniform boundedness property of , crucial to guarantee optimal error estimates [9], is valid in multiple cases (see Chapters 4 and 5 in [9] or [12]), but not systematically, (see for example first-order PDE’s in [9]).
Next, from (11) we obtain:

(12)

where .
In the sequel, we will denote by the following expression:

(13)

Finally, we also observe that there exists a critical value of defined by:

(14)

such that the error estimate (12) can be splitted according to:

(15)

Based on this error estimate, the following section is devoted to derive a probabilistic model applied to the relative accuracy between two Lagrange finite elements and when the mesh size has a given and fixed value.

3 The probabilistic analysis of the relative finite elements accuracy

In [6] we proposed two probability distributions which enabled us to appreciate the evaluation of the more likely accurate between two Lagrange finite elements and . These distributions were essentially derived by considering the error estimate (10).
In the present paper we generalize these probability distributions by introducing two new inputs which are:

  1. The error estimate (10) is enriched by the a priori estimates (2) and (4) to finally consider (15).

  2. The probabilistic approach we develop is an extension of those we considered in [4] and [6]. More precisely, by the help of an ad-hoc density probability, we derive the probability distribution of a suitable random variable so that we can compare the two approximation errors and , considered as random variables, as will be introduced now.

3.1 Random solution and random approximations of determinist partial differential equation

The purpose of this section is based of the following fundamental remark: Solution to the variational problem (VP), except for particular cases, is totally unknown (being impossible to calculate it analytically); this motivates the numerical schemes one will choose to implement.
This inability to determine, in most cases, the exact solution , is mainly due to the complexity of the involved PDE’s operator; it indeed depends on complex combinations of integrals, partial derivatives and boundary conditions, as well as on the bent geometrical shape of the domain of integration . All of these ingredients hence participate in the incapability to analytically determine the exact solution as their relationship with is inextricable , and then, unknown.
As a consequence, this lack of knowledge and information regarding the dependency between these ingredients and the solution motivates us to consider as a random variable, as well as any function of . This paper is dedicated to the approximation error of , considered as a random variable.
In this frame, we view solution to the variational formulation (VP) defined by (1) in the same way as it is usual to consider the trajectory and the contact point with the ground of any solid body which is thrown, i.e., as random. Indeed, in this case, due to the lack of information concerning the initial conditions of the trajectory of the body, the solution of the concerned inverse kinematic operator is inaccessible, and is thus seen as a random variable.
In the case analyzed in this paper, the situation is much worse. Indeed, we investigate a general variational formulation (VP) where the analog of the inverse kinematic operator is too complex to enable us to analytically determine the corresponding solution of (VP) by any mathematical expression. It is one of the reasons which motivate us to view solution and its approximation as random variables, since the corresponding approximate operator conserves the complexity of the original one described above.

3.2 The probabilistic distribution of the relative finite elements accuracy

In the previous section we motivated the reason why we consider solution and its approximation as a random variables.
To complete the description of the randomness feature of the approximation error , we also remark that, since the way a given mesh grid generator will produce any mesh is random, then the corresponding approximation is random too.
For all of these reasons, based on the error estimate (12)-(13), we can only affirm that the value of the approximation error is somewhere within the interval .
As a consequence, we decide to see norm as a random variable defined as follows:
Let and be fixed. We introduce the random variable defined by:

(16)
(17)

Thus, the space product plays the role of the usual probability space introduced in this context.
Now, regarding the absence of information concerning the more likely or less likely values of norm within the interval , we will assume that the random variable

is uniformly distributed over the interval

, with the following meaning:

(18)

Equation (18) means that if one slides the interval anywhere in , the probability of event does not depend on where the interval is located in , but only on its length; this corresponds to the property of uniformity of the random variable .
Let us now consider two families of Lagrange finite elements and corresponding to a set of values such that .
The two corresponding inequalities given by (12)-(13), assuming that solution to (VP) belong to , are:

(19)
(20)

where and respectively denote the and Lagrange finite element approximations of and as defined by (13).

Remark 2

If one considers a given mesh for the finite element which contains that of then, for the particular class of problems where (VP) is equivalent to a minimization formulation (MP) (see for example [3]), one can show that the approximation error of the finite element is always smaller than that of , and is more accurate than , for all values of the mesh size .

Therefore, to avoid this situation, for a given value of , we consider two independent meshes built by a mesh generator for and . Now, usually, in order to compare the relative accuracy between these two finite elements, one asymptotically considers inequalities (19) and (20) to conclude that, when goes to zero, is more accurate that , since goes faster to zero than .
However, for a given application, has a given and fixed value, so this way of comparison is not valid anymore. For this reason, our purpose is to determine the relative accuracy between two finite elements and for a fixed value of corresponding to two independent meshes.
Moreover, since we chose to consider the two random variables as uniformly distributed on their respective interval of values , we also assume that they are independent. This assumption is, once again, the result of the lack of information which lead us to model the relationship between these two variables as independent, since any knowledge is available to more precisely localize the value of if the value of was known, and vice versa.
By the following result, we establish the density of probability of the random variable defined by: .

Theorem 3.1

Let be the two uniform and independent random variables defined by (16)-(17): , where is defined by (13).
Then, the random variable defined on has the following density of probability:

Proof : Let us now remark that, since the support of the two random variables is , the support of the density is therefore , which corresponds to (21).
Let us consider the case where .
If denotes the density of probability defined on associated to each random variable , since we assume they are independent variables, the density is given by:

(26)

where:

(27)

and is the indicator function of the interval .
Furthermore, due to the definition (27) of the density for each variable the integrand of (26) can be expressed as follows:

As a consequence, , which leads one to consider the five following cases corresponding to the significant relative positions between the intervals and .

  1. Let us assume that . If then and which is again the result of (21).

  2. We consider now the values of such that and .
    If , then and by (26) we get the expected expression of (25) since we have:

    (30)
  3. The following case concerns the values of such that and . If , then and we get (23) since:

    (31)
  4. If and then which is impossible since .

  5. We consider now the values of such that and .
    If , then and, since we have:

    (32)

    we get the expected expression of (22).

  6. Finally, if , then and which corresponds to (21).

The other cases corresponding to can be deduced using the same arguments.  

From Theorem 3.1

, one can infer the entire cumulative distribution function

defined by:

(33)

However, since we are interested in determining the more likely finite element between and , we focus the following corrolary to the value of which corresponds to .

Corollary 3.2

Let be the two uniform and independent random variables defined by (16)-(17): , where is defined by (13). Then, we have:

Proof :

  • Let us consider the case where . Then, by definition (33) of the entire cumulative distribution function , at , we have:

    (36)

    where we used the values of the density given by (22) and (23).

  • In the same way, when we have:

    (37)

 

We can now explicit the probability distribution of the event given by (34)-(35) as a function of the mesh size .
To this end, we remark that, since each has two possible values depending on the relative position between and defined by (14), the probability distribution we are looking for must be splitted in the corresponding cases as well.
Let us hence introduce the constants defined by:

(38)

and the specific value of which corresponds to the intersection of the curves defined by , (see Figure 1).
Then, we have:

(39)

We notice that and strongly depend on and , since and depend on these two parameters as well. As a consequence, in the following theorem the different formulas of will contain this dependency on and .

Theorem 3.3

Let be the two uniform and independent random variables defined by (16)-(17). Then, is determined by:
If then and:

If then and:

Proof : To establish the proof of Theorem 3.3, we will consider a geometrical interpretation of the error estimate (19) and (20).
These two inequalities can indeed be geometrically viewed with the help of the relative position between the two curves introduced above, and the horizontal line defined by