ABSTRACT
In this paper, we describe an a posteriori error analysis for a conforming dual mixed scheme of the Poisson problem with non homogeneous Dirichlet boundary condition. As a result, we obtain an a posteriori error estimator, which is proven to be reliable and locally efficient with respect to the usual norm on . We remark that the analysis relies on the standard Ritz projection of the error, and take into account a kind of a quasi-Helmholtz decomposition of functions in , which we have established in this work. Finally, we present one numerical example that validates the well behavior of our estimator, being able to identify the numerical singularities when they exist.
Keywords:
mixed finite element methods; a posteriori error estimator; reliability; efficiency
1 INTRODUCTION
It is well known that when the solution of a variational formulation obtained by applying a finite element method, is not smooth enough, the quality of approximation could be not good enough. This motivates us to derive an a posteriori error estimator, which is reliable and efficiency. This would allow us to establish that the estimator behaves as the error of the method, which in general is not known. Then, considering an appropriate adaptive refinement algorithm, we can obtain approximations of the formulation, of better quality, by detecting the region where this estimator is more dominant. In the context of mixed finite element methods, there are a lot of references dedicated to the a posteriori error analysis. For instance, in 11 A. Alonso. Error estimators for a mixed method. Numer. Math., 74(4) (1996), 385-395. doi:10.1007/s002110050222. URL https://doi.org/10.1007/s002110050222.
https://doi.org/10.1007/s002110050222...
an a posteriori error estimator only for the flux unknown is derived, using Raviart-Thomas (RT) or Brezzi-Douglas-Marini (BDM) as its space of approximation. The analysis that yields this estimator, relies on a classical Helmholtz decomposition. On the other hand, in 1010 D. Braess & R. Verfürth. A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal., 33(6) (1996), 2431-2444. doi:10.1137/S0036142994264079. URL https://doi.org/10.1137/S0036142994264079.
https://doi.org/10.1137/S003614299426407...
, the authors present two a posteriori error estimators for a dual mixed formulation for the Poisson problem, approximating the flux in the Raviart-Thomas space. In this case, the derivation of the estimator is obtained under a saturation assumption. This requirement is circumvented in 1212 C. Carstensen. A posteriori error estimate for the mixed finite element method. Math. Comp., 66(218) (1997), 465-476. doi:10.1090/S0025-5718-97-00837-5. URL https://doi.org/10.1090/S0025-5718-97-00837-5.
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, where a reliable and efficient a posteriori error estimator for the natural norm, is derived. We remark that four different kind of a posteriori error estimators for Raviart-Thomas mixed finite elements, are provided in 2323 B.I. Wohlmuth & R.H.W. Hoppe. A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements. Math. Comp. , 68(228) (1999), 1347-1378. doi:10.1090/S0025-5718-99-01125-4. URL https://doi.org/10.1090/S0025-5718-99-01125-4.
https://doi.org/10.1090/S0025-5718-99-01...
. Concerning second order elliptic equation with mixed boundary condition, in 1717 G.N. Gatica & M. Maischak. A posteriori error estimates for the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differential Equations, 21(3) (2005), 421-450. doi:10.1002/num.20050. URL https://doi.org/10.1002/num.20050.
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the authors developed an a posterior error analysis for the mixed finite element method with a Lagrange multiplier.
In 99 T.P. Barrios & G.N. Gatica. An augmented mixed finite element method with Lagrange multipliers: A priori and a posteriori error analyses. J. Comput. Appl. Math. , 200(2) (2007), 653-676. doi:10.1016/j.cam.2006.01.017. URL https://doi.org/10.1016/j.cam.2006.01.017.
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, an a posteriori error analysis for an augmented mixed formulation of the Poisson problem with mixed boundary conditions, is developed. This is performed with the help of the Ritz projection of the error, and covers the reliability and efficiency of the estimator. It is important to remark that this technique has been successfully applied to other problems, such as the the Brinkman model in 22 T. Barrios, R. Bustinza, G.C. García & M. González. An a posteriori error estimator for a new stabilized formulation of the Brinkman problem. In “Numerical mathematics and advanced applications- ENUMATH 2013”, volume 103 of Lect. Notes Comput. Sci. Eng. Springer, Cham (2015), p. 253-261., the Darcy flow in 66 T.P. Barrios, J.M. Cascón & M. González. A posteriori error analysis of an augmented mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Engrg., 283 (2015), 909-922. doi:10.1016/j.cma.2014.10.035. URL https://doi.org/10.1016/j.cma.2014.10.035.
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and 77 T.P. Barrios, J.M. Cascón & M. González. A posteriori error estimation of a stabilized mixed finite element method for Darcy flow. In “Boundary and interior layers, computational and asymptotic methods-BAIL 2014”, volume 108 of Lect. Notes Comput. Sci. Eng. Springer, Cham (2015), p. 13-23., the Stokes system in 33 T.P. Barrios, E.M. Behrens & R. Bustinza. A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: the stationary case. Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday. Internat. J. Numer. Methods Fluids, 92(6) (2020), 509-527. doi:10.1002/fld.4793. URL https://doi.org/10.1002/fld.4793.
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and 55 T.P. Barrios, R. Bustinza, G.C. García & M. González. An a posteriori error analysis of a velocity- pseudostress formulation of the generalized Stokes problem. J. Comput. Appl. Math., 357 (2019), 349-365. doi:10.1016/j.cam.2019.02.019. URL https://doi.org/10.1016/j.cam.2019.02.019.
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, and the Oseen equations in 88 T.P. Barrios, J.M. Cascón & M. González. Augmented mixed finite element method for the Oseen problem: A priori and a posteriori error analysis. Comput. Methods Appl. Mech. Engrg. , 313 (2017), 216-238., for example.
In this paper, we deduce a reliable and efficient residual a posteriori error estimator for the Poisson problem with non homogeneous Dirichlet boundary condition, considering a dual mixed finite element method. To achieve this, we take into account the Ritz projection of the error, measured in the standard norm. We also establish another kind of quasi Helmholtz decomposition of in the plane. We remark that in this process, no saturation assumption is required, and its extension to 3D case is not difficult. We remark that in 11 A. Alonso. Error estimators for a mixed method. Numer. Math., 74(4) (1996), 385-395. doi:10.1007/s002110050222. URL https://doi.org/10.1007/s002110050222.
https://doi.org/10.1007/s002110050222...
the a posteriori error analysis is performed to a homogeneous Dirichlet problem, focusing in obtain an estimator for the norm of the flux error. Then, the results of the current work can be seen as a natural extension of what is described in 11 A. Alonso. Error estimators for a mixed method. Numer. Math., 74(4) (1996), 385-395. doi:10.1007/s002110050222. URL https://doi.org/10.1007/s002110050222.
https://doi.org/10.1007/s002110050222...
, since we deduce an a posteriori error estimator for the norm of the error of the flux and potential unknowns, that is reliable and efficient.
The rest of the article is organized as follows: In Section 2 we present the model problem, as well as the corresponding dual mixed formulations, at continuous and discrete levels. Next, the a posteriori error analysis with non homogeneous Dirichlet is described in Sections 3. This includes the introduction of the Ritz projection of the error, as well as the key tool for deducing a reliable a posteriori error estimator: a quasi-Helmholtz decomposition of functions in . Finally, one numerical example confirming our theoretical results are reported in Section 4. We end this introduction with some notation to be used throughout the paper. Given any Hilbert space H, we denote by H 2 the space of vectors of order 2 with entries in H. Finally, we use C or c, with or without subscripts, to denote generic constants, independent of the discretization parameter, that may take different values at different occurrences.
2 MODEL PROBLEM AND VARIATIONAL FORMULATIONS
Let Ω be a bounded and simply connected domain in ℝ2 with polygonal boundary Γ. Then, given and , we consider the model problem: Find such that −∆u = f in Ω and u = g on Γ. Since we are interested in dual mixed methods, we rewrite the Dirichlet problem as the first order system: Find (σ, u ) such that σ = −∇u in Ω, div(σ ) = f in Ω, and u = g on Γ. Hence, proceeding in the usual way, we arrive to the following dual mixed variational formulation: Find such that
where denotes the duality pairing between H -1/2(Γ) and H 1/2(Γ) with respect to L 2(Γ)- inner product, and the bilinear forms and , are given by and , respectively. Thanks to the classical Babuška-Brezzi theory (cf. Section 5 in 1616 R.G. Durán. Mixed Finite Element Methods. In “Mixed finite elements, compatibility conditions, and applications”, Lecture Notes in Mathematics. Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence (2008), p. 1-44.), it can be shown that there exists a unique pair solution of (2.1). For the discretization, we assume that Ω is a polygonal region and let be a regular family of triangulations of such that . For any triangle , we denote by h T its diameter and define the mesh size . In addition, given an integer ℓ ≥ 0 and a subset S of ℝ2, we denote by 𝒫ℓ(S) the space of polynomials in two variables defined in S of total degree at most ℓ, and for each , we define the local Raviart-Thomas space of order κ ≥ 0 (cf. 2020 J.E. Roberts & J.M. Thomas. Mixed and hybrid methods. In “Handbook of numerical analysis, Vol. II”, Handb. Numer. Anal., II. North-Holland, Amsterdam (1991), p. 523-639.), . Then, given an integer r ≥ 0, we define the finite element subspaces and . Under these assumptions, and applying a discrete version of the Babuška-Brezzi theory (see Section 5 in 1616 R.G. Durán. Mixed Finite Element Methods. In “Mixed finite elements, compatibility conditions, and applications”, Lecture Notes in Mathematics. Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence (2008), p. 1-44.), we can ensure that there exists only one such that
Moreover, the following result is established.
Theorem 2.1.Let (σ, u ) and (σ h , u h ) be the solutions of (2.1) and (2.2), respectively. If, and, then there exists C > 0, independent of the mesh size, such that
Proof. We refer to the proofs of Theorems 3.2 and 3.3 in 1616 R.G. Durán. Mixed Finite Element Methods. In “Mixed finite elements, compatibility conditions, and applications”, Lecture Notes in Mathematics. Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence (2008), p. 1-44., as well as the classical error estimates for the L 2-orthogonal projection onto 𝒫r . We omit further details. □
3 A POSTERIORI ERROR ANALYSIS
In this section, we follow 44 T.P. Barrios, E.M. Behrens & M. González. Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity. Appl. Numer. Math. , 84 (2014), 46-65. doi:10.1016/j.apnum.2014.05.008. URL https://doi.org/10.1016/j.apnum.2014.05.008.
https://doi.org/10.1016/j.apnum.2014.05....
(see also 55 T.P. Barrios, R. Bustinza, G.C. García & M. González. An a posteriori error analysis of a velocity- pseudostress formulation of the generalized Stokes problem. J. Comput. Appl. Math., 357 (2019), 349-365. doi:10.1016/j.cam.2019.02.019. URL https://doi.org/10.1016/j.cam.2019.02.019.
https://doi.org/10.1016/j.cam.2019.02.01...
), and develop an a posteriori error analysis for the discrete scheme (2.2), taking into account an appropriate Ritz projection of the error and a quasi- Helmholtz decomposition. We first introduce some notations and results, concerning the Clément and Raviart-Thomas interpolation operators.
3.1 Notation and some well known results
Given , we let E(T) be the set of its edges. By E h we denote the set of all edges (counted once) induced by the triangulation 𝒯h . Then, we write , where and . Similarly, N h will denote the list of all vertices (counted once) induced by the triangulation 𝒯h . Then we define and . As a result, we have that . In addition, for each , and for each . Now, given and , we set
Also, for each , we fix a unit normal exterior vector , and let be the corresponding fixed unit tangential vector along ∂T. From now on, when no confusion arises, we simply write n and t instead of n T and t T , respectively. In addition, let q and τ be scalar - and vector -valued functions, respectively, that are smooth inside each element . We denote by (q T,e , τ T,e ) the restriction of (q T , τ T ) to e. Then, given , we define the jump of q and of the tangential component of τ at x ∈ e, by
where T and T’ are the two elements in 𝒯h sharing the edge . On boundary edges , we set , where is such that . Finally, given a smooth scalar field v and a vector field , we define
Next, we introduce the Clément interpolation operator (cf. 1515 P. Clément. Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér., 9(R-2) (1975), 77-84.), where . The following lemma establishes the main local approximation properties of I h .
Lemma 3.1.There exist constants c1, c2 > 0, independent of h, such that for all, there holds
and
where hedenotes the length of the side.
Proof. We refer to 1515 P. Clément. Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér., 9(R-2) (1975), 77-84.. □
On the other hand, we also need to introduce the Raviart-Thomas interpolation operator (see 1111 F. Brezzi & M. Fortin. “Mixed and hybrid finite element methods”, volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991), x+350 p. doi:10.1007/978-1-4612-3172-1. URL https://doi.org/10.1007/978-1-4612-3172-1.
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, 2020 J.E. Roberts & J.M. Thomas. Mixed and hybrid methods. In “Handbook of numerical analysis, Vol. II”, Handb. Numer. Anal., II. North-Holland, Amsterdam (1991), p. 523-639.), , which given is characterized by the following conditions:
and
The operator satisfies the following approximation properties.
Lemma 3.2.There exist constants c3, c4, c5> 0, independent of h, such that for all
for all with ,
and for any
where, such that it contains e on its boundary.Proof. See e.g. 1111 F. Brezzi & M. Fortin. “Mixed and hybrid finite element methods”, volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991), x+350 p. doi:10.1007/978-1-4612-3172-1. URL https://doi.org/10.1007/978-1-4612-3172-1.
https://doi.org/10.1007/978-1-4612-3172-...
or 2020 J.E. Roberts & J.M. Thomas. Mixed and hybrid methods. In “Handbook of numerical analysis, Vol. II”, Handb. Numer. Anal., II. North-Holland, Amsterdam (1991), p. 523-639.. □
In addition, the interpolation operator can also be defined as a bounded linear operator from the larger space into , for all (see, e.g. Theorem 3.16 in 1919 R. Hiptmair. Finite elements in computational electromagnetism. Acta Numer., 11 (2002), 237-339. doi:10.1017/S0962492902000041. URL https://doi.org/10.1017/S0962492902000041.
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). In this case, there holds the following interpolation error estimate
Taking into account (3.1) and (3.2), it is not difficult to show that
where is the L2 −orthogonal projector. On the other hand, it is well known (see, e.g. 1414 P.G. Ciarlet. “The finite element method for elliptic problems”, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002), xxviii+530 p. doi:10.1137/1.9780898719208. URL https://doi.org/10.1137/1.9780898719208. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
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) that for each , with , there exists C > 0, independent of h, such that
3.2 Reliability of the estimator
Let and be the unique solution to problems (2.1) and (2.2), respectively. We provide Σ with its usual inner product
which induces the norm
Next, we consider the Ritz projection of the error with respect to as the unique element , such that
where the global bilinear form arises from the variational formulation (2.1), after adding its equations, that is
We remark that the existence and uniqueness of is guaranteed by the Lax-Milgram Lemma. Moreover, we point out that the properties of the bilinear forms a(·,·) and b(·,·) implies that A(·,·) satisfies a global inf-sup condition, i.e., there exist α > 0 such that
This particularity allows us to bound the error in terms of the solution of its Ritz projection, as follows:
Then, according to (3.9), and with the purpose of obtaining a reliable a posteriori error estimate for the discrete scheme (2.2), it is enough to bound from above the Ritz projection of the error. To this aim, the next result will be useful, and can be seen as a kind of a quasi-Helmholtz decomposition of functions in H(div; Ω).
Lemma 3.3.For each, there existand, such that
where (a, b)t is any fixed point belonging to Ω, and. In addition, there exists C > 0, such that
Proof. We first introduce the space . Next, for each , we decompose , where .
We remark that . Then, since , and invoking Corollary I.2.4 in 1818 V. Girault & P.A. Raviart. “Finite element methods for Navier-Stokes equations”, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986), x+374 p. doi:10.1007/978-3-642-61623-5. URL https://doi.org/10.1007/978-3-642-61623-5. Theory and algorithms.
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, there exists such that in Ω and . This implies that
where (a, b)t is a fixed point belonging to Ω. Hence, by Theorem I.3.1 in 1818 V. Girault & P.A. Raviart. “Finite element methods for Navier-Stokes equations”, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986), x+374 p. doi:10.1007/978-3-642-61623-5. URL https://doi.org/10.1007/978-3-642-61623-5. Theory and algorithms.
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, there exists a stream function such that in Ω. In addition, we have
As a result, we establish (3.11), and we end the proof. □
Now, considering χ and Φ as the ones provided by Lemma 3.3 for a given , we introduce , and define
which is referred as a discrete quasi-Helmholtz decomposition of τ h . Therefore, we can write
that verifies
On the other hand, it is not difficult to check the following orthogonality relation
From now on, given , we associate it with the discrete pair , where τ h is defined as in (3.12). Hence, considering (3.15) with , and knowing that (σ, u ) is the unique solution of problem (2.1), we obtain
Equivalently,
where and are the bounded linear functionals defined by
Hence, taking into account (3.13) and (3.14), and the fact that on Γ, we can rewrite F 1(τ − τ h ) as follows
where
Our aim now is to obtain upper bounds for each one of the terms F 2(v), R 1(Φ) and R 2(χ).
Lemma 3.4. For any there holds
Proof. The proof follows from a straightforward application of Cauchy-Schwarz inequality. □
Lemma 3.5.There exists C > 0, independent of h, such that
Proof. It is a slight modification of Lemma 3.5 in 55 T.P. Barrios, R. Bustinza, G.C. García & M. González. An a posteriori error analysis of a velocity- pseudostress formulation of the generalized Stokes problem. J. Comput. Appl. Math., 357 (2019), 349-365. doi:10.1016/j.cam.2019.02.019. URL https://doi.org/10.1016/j.cam.2019.02.019.
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. We omit further details. □
Lemma 3.6.Under the assumption that, there exists C > 0, independent of h, such that
Proof. Knowing that on Γ, and after integrating by parts, we deduce
Therefore, the proof is completed invoking Lemma 3.1, the Cauchy-Schwarz inequality, the regularity of the mesh and (3.11). □
The previous results suggest the definition of the following residual estimator
where
An upper bound for is established in the next lemma, in terms of (3.16).
Lemma 3.7.Assuming that, there exists a constant C > 0, independent of h, such that
Proof. Invoking Lemmas 3.5 and 3.6, we deduce that there exists C > 0, independent of h, such that
Hence, (3.17) follows from the above bound, Lemma 3.4 and a discrete Cauchy-Schwarz inequality. □
The following theorem establishes the main result of this section, which is the reliability and efficiency of the estimator η.
Theorem 3.2.There exists a positive constant Crel, independent of h, such that
Additionally, there exists Ceff > 0, independent of h, such that
where . Proof. The reliability of η, (3.18), follows from (3.9) and Lemma 3.7. The efficiency of η, (3.19), is treated in the next subsection. We omit further details. □
3.3 Efficiency of the estimator
In this subsection we prove the local efficiency of the estimator η (cf. (3.19)). We begin by introducing some notations and preliminary results. Given and , we let ψ T and ψ e be the standard triangle-bubble and edge-bubble functions, respectively. In particular, ψ T satisfies on ∂T, and in T. Similarly, on ∂ω e , and in ω e . We also recall from 2121 R. Verfürth. A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. , 50(1-3) (1994), 67-83. doi:10.1016/0377-0427(94)90290-9. URL https://doi.org/10.1016/0377-0427(94)90290-9.
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that, given , there exists an extension operator that satisfies and . Additional properties of ψ T , ψ e , and L are collected in the following lemma.
Lemma 3.8. For any triangle T there exist positive constants c 1 , c 2 , c 3 and c 4 , depending only on k and the shape of T, such that for all and , there hold
Proof. See Lemma 4.1 in 2121 R. Verfürth. A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. , 50(1-3) (1994), 67-83. doi:10.1016/0377-0427(94)90290-9. URL https://doi.org/10.1016/0377-0427(94)90290-9.
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. □
The following inverse estimate will also be useful.
Lemma 3.9.Letsuch that ℓ ≤ m. Then, for any triangle T , there exists c > 0, depending only on k, ℓ, m and the shape of T, such that
Proof. See Theorem 3.2.6 in 1414 P.G. Ciarlet. “The finite element method for elliptic problems”, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002), xxviii+530 p. doi:10.1137/1.9780898719208. URL https://doi.org/10.1137/1.9780898719208. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
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. □
Since f = div(σ ) in Ω, we have that
Lemma 3.10.There exists C1> 0, independent of the meshsize, such that for any
Proof. We introduce in T. Then, taking into account the property (3.20) and integrating by parts, we have
Now, applying Cauchy-Schwarz inequality as well as inverse inequality (3.23) and property 0 ≤ ψ T ≤ 1, we derive
Hence, simplifying and multiplying by the factor h T , we complete the proof of the lemma. □
In the following lemma, we bound the jump of u h ,
Lemma 3.11.There exists C2> 0, independent of the mesh size, such that for any
Proof. First, given we set , with such that . Next, we introduce on e and in ω e , which belongs to H(div, ω e ). Taking into account (3.21), knowing that [[u]]= 0 on E I , and integrating by parts, we derive
Using the fact that , and applying Cauchy-Schwarz inequality, we deduce
Now, invoking the inverse inequality (3.23) and knowing that 0 ≤ ψ e ≤ 1 in ω e together with (3.22), we arrive for each
This inequality, together with (3.22), allow us to rewrite (3.25) as follows: There exists c > 0 independent of mesh size, such that
Then the proof follows after multiplying by h e , and applying Lemma 3.10. □
Lemma 3.12.There exists C3 > 0, independent of the meshsize, such that for any
Proof. We introduce in T. Then, invoking the property (3.20), rot(σ) = 0 in T, and integrating by parts, we have
Now, applying Cauchy-Schwarz inequality, as well as inverse inequality (3.23) and the fact that 0 ≤ ψT ≤ 1 in T, we derive
Hence, simplifying and multiplying by the factor h T , we complete the proof of the lemma. □
The tangential component jump of σ h is treated in the next lemma.
Lemma 3.13.There exists C4 > 0, independent of the mesh size, such that for any
Proof. Given , let such that and they share e, i.e. . Denoting by on e, and using 3.21, it follows that
where in the last equality we take into account
In addition, realizing that and applying Cauchy-Schwarz inequality, we deduce
Now, knowing that , and taking into account (3.22), for each , we deduce
Now, the inverse inequality (3.23)), the fact that in ωe , together with (3.22)), implies for each
Inequalities (3.29) and (3.30) allow us to rewrite (3.28) as follows: There exists c > 0 independent of meshsize, such that
Then, (3.26) follows after simplifying , multiplying by and invoking Lemma 3.12. □
Remark 3.14.The current a posteriori error analysis can be extended to three dimensions. To this aim, we consider Ω a bounded and simply connected polyhedral domain in ℝ3 . Now, given a partition 𝒯h ofmade of tetrahedral, we take into account similar notations as the ones introduced in Section 3, with face instead of edge. In addition, for any smooth enough vector field ρ, respectively, we set, while the jump of tangential trace of ρ across, by
whereare the pair of tetrahedral sharing the face. On the other hand, when, bywe refer to the unique element having e as a boundary face. Now, following the ideas given in the proof of Lemma 3.3, and applying Theorem I.3.5 in1818 V. Girault & P.A. Raviart. “Finite element methods for Navier-Stokes equations”, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986), x+374 p. doi:10.1007/978-3-642-61623-5. URL https://doi.org/10.1007/978-3-642-61623-5. Theory and algorithms.
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, we can establish the 3D-version of the quasi-Helmholtz decomposition of functions belonging to H(div; Ω) presented in Lemma 3.3, which in addition is also stable (invoking Theorem 2.1 in1313 C. Carstensen, S. Bartels & S. Jansche. A posteriori error estimates for nonconforming finite element methods. Numer. Math. , 92(2) (2002), 233-256. doi:10.1007/s002110100378. URL https://doi.org/10.1007/s002110100378.
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). This means that for any, there existand, such that
where (a, b, c)t is any fixed point belonging to Ω, and. In addition, there exists C > 0, such that
Then, proceeding in analogous way as in Section 3, we prove a similar result to Theorem 3.2, where the local a posteriori error estimator now reads as
4 NUMERICAL EXAMPLE
In this section, we present one numerical example illustrating the performance of the dual mixed method when applied to the Poisson problem, with Dirichlet condition, as well as of the corresponding adaptive procedure. We consider the lowest finite element for our approximation. We remark that the computational implementation has been done using a MATLAB code.
Hereafter, the number of degrees of freedom (unknowns) is given by N: = number of edges + number of elements, induced by the triangulation. Moreover, the involved individual and total errors are defined as and , where and are the corresponding unique solutions of the continuous (2.1) and discrete (2.2) formulations. Additionally, if e and e’ stand for the errors at two consecutive triangulations with N and N’ number of degrees of freedom, respectively, we set the experimental rate of convergence of the global error as . We define r0(u) and r(σ ) in analogous way.
The data f and g for our example, are chosen so that the exact solution is and . We notice that in this case u has a singularity at (−1.05, 0), which does not belong to Ω, but it is very close to ∂Ω. Then, u has a numerical singularity in a neighborhood of .
Then, the purpose of this example, is to show the performance of the following adaptive algorithm (cf. 2222 R. Verfürth. “A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques”. Advances in Numerical Mathematics. Wiley-Teubner, Chichester (1996), xx+150 p.). Given an a posteriori error estimator:
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1. Start with a coarse mesh 𝒯h .
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2. Solve the Galerkin scheme for the current mesh 𝒯h .
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3. Compute ηT for each triangle T ∈ 𝒯h .
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4. Consider stopping criterion and decide to finish or go to the next step.
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5. Apply Blue-green procedure to refine each element T’ ∈ 𝒯h such that
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6. Define the resulting mesh as the new 𝒯h and go to step 2.
Table 1 reports the histories of convergence of the individual and total errors for a sequence of uniform and adaptive refinements, respectively. We notice that the adaptive refinement algorithm is able to recognize the numerical singularity, and then the induced sequence of adapted meshes let us to improve the quality of approximation, better than the corresponding when uniform refinement is performed. In addition, we observe that index of efficiency e/η remains bounded, indicating that η is reliable and efficient, despite the fact that g in this case is not piecewise polynomial. Figure 1 displays some adapted meshes, generated by the proposed adaptive algorithm, from which we observe that the numerical singularity is detected.
History of convergence of Example provided, considering uniform (up) and adaptive (bottom) refinements.
Adapted meshes corresponding (top-bottom, left-right) to 192, 386, 3633 and 29052 dofs, for Example considered, with Dirichlet boundary condition (based on η).
CONCLUDING REMARKS
In this paper, we have developed an a posteriori error analysis for a dual mixed formulation of Poisson problem in the plane, with non homogeneous Dirichlet boundary condition. By establishing a new kind of quasi-Helmholtz decomposition of functions in H(div; Ω) (cf. Lemma 3.3), we are able to obtain an a posteriori error estimator, which consists of six residual terms, and results to be reliable and locally efficient with respect to the error measured in its natural norm on . In this sense, we have generalized the results obtained in previous works ( 11 A. Alonso. Error estimators for a mixed method. Numer. Math., 74(4) (1996), 385-395. doi:10.1007/s002110050222. URL https://doi.org/10.1007/s002110050222.
https://doi.org/10.1007/s002110050222...
, 1010 D. Braess & R. Verfürth. A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal., 33(6) (1996), 2431-2444. doi:10.1137/S0036142994264079. URL https://doi.org/10.1137/S0036142994264079.
https://doi.org/10.1137/S003614299426407...
, for example), and without invoking the so called saturation assumption.
The results of numerical experiment, included in this work, are in agreement with our theoretical analysis. Here, we notice that the estimator is able to help us to identify which part of the domain is localized the numerical singularity of the exact solution. As a consequence, the adaptive algorithm, based on this estimator, let us to improve the quality of the approximation.
Finally, since Lemma 3.3 can be proved for 3d case too, the current work can be extended to 3d, obtaining a reliable and locally efficient residual a posteriori error estimator, consisting also of six residual terms (cf. (3.31)).
Acknowledgments
This research was partially supported by ANID-Chile through the the project Centro de Modelamiento Matemático (AFB170001) of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal CONICYT-Chile and FONDECYT grant No. 1200051; by Direcciones de Investigación y de Postgrado de la Universidad Católica de la Santísima Concepción (Chile), through Incentivo Mensual and Becas de Mantención programs, and by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción (Chile).
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Publication Dates
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Publication in this collection
05 Sept 2022 -
Date of issue
Jul-Sep 2022
History
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Received
27 Sept 2021 -
Accepted
24 Mar 2022