A Posteriori Error Estimates of Residual Type for Second Order Quasi-Linear Elliptic PDEs

where Ω is assumed to be a polygonal bounded domain in R2, f ∈ L2(Ω), and α is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the H1-norm by an indicator η which is composed of L2norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the α-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.


Introduction
A posteriori error estimation began playing role in analyzing the accuracy of the numerical solution with a pioneering work of Babuška and Rheinboldt (Babuška, I., 1978).A local estimator not only shows us how good the approximation performs, but sometimes also acts as an indicator used to determine whether that local mesh should be refined.From this usage, a new mesh will be created with the expectation that it will improve the accuracy of the approximation in efficiency way, without increasing the degree of polynomials used in the approximation.All of this ensemble forms the following procedure: In principle, the local estimator, or indicator, should be derived elementwise from the problem residual which is computed from the discrete solution and the given data of the problem.Thus, after solving, the indicators will be output from the submodule ESTIMATE.All elements with higher value of the indicators than the user's tolerance must be marked.Those marked elements then must be divided by some appropriate strategies.The new discrete problem with the resulted finer mesh is now ready to be solved again.The adaptive algorithm iterates the above procedure until the overall error is determined small enough.Applying finite element method in the step SOLVE allows us to call this process as the adaptive finite element method (AFEM).
The introductory principles of adaptive finite elements and additional references can be found in the books by Ainsworth and Oden (Ainsworth, M., 2000), and Verfürth (Verfürth, R., 1996).For the linear case we refer to the works of Morin, Nochetto, and Siebert (Morin, P., 2000), where the convergence for second order elliptic equations with piecewise constant coefficients and without lower order terms were investigated by using a technique originated by Dörfler (Dörfler, W., 1996).They also introduced the notion of data oscillation meant to quantify information missed in projecting the residual with discrete functions which is a process associated with the finite element method.Thereafter, Mekchay and Nochetto extended these results for general second order linear elliptic PDEs (Mekchay, K., 2005), and (in cooperation with Morin) for the Laplace-Beltrami operator on surfaces (Mekchay, K., 2011).Recently, Garau, Morin, and Zuppa (Garau, E. M., 2011) designed an adaptive finite element algorithm for solving quasi-linear elliptic problems based on a Kačanov iteration.They estimated the problem residual instead of the actual error, which need a practical way to deal with the negative norm in the dual space H −1 .The quasi-optimal convergence rate of the algorithm were proved in (Garau, E. M., in press).
The objective of this article is to obtain a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator: where Ω is assumed to be a polygonal bounded domain in R 2 , f ∈ L 2 (Ω), and α is a bounded function which satisfies the monotonic properties (see assumptions ( 3)-( 4) below) for admission of a unique weak solution.We estimated the actual error in the H 1 -norm by an indicator η which is composed of L 2 -norms of the element residual and the jump residual.
This paper is organized as follows.In §2, we give the weak formulation of (1) and its corresponding discrete problem, together with some assumptions imposed to α for admission of a unique weak solution.The analysis of a posteriori error estimation is described in §3, which is divided into two parts; the upper bound and the lower bound for the error.Then we discuss about the adaptive algorithm in the last section.

Problem Formulations
By L 2 (Ω), we denote the usual Lebesgue space with norm The Sobolev space of functions u ∈ L 2 (Ω) with weak derivatives ∇u ∈ L 2 (Ω) is denoted by H 1 (Ω) with semi norm and norm Normally, ∥u∥ 0,ω and ∥u∥ 1,ω represent L 2 -norm and H 1 -norm restricted on the subdomain ω, respectively.According to the boundary condition of Dirichlet type, H 1 0 (Ω) is a subset of H 1 (Ω) composed of functions vanishing on ∂Ω.We multiply the PDE (1) by a smooth test function ϕ ∈ C ∞ (Ω) and integrate by parts over Ω to admit the weak formulation: where According to the monotonicity methods described in (Evans, L. C., 1998), to guarantee the existence of a unique weak solution of (2), the vector field ⃗ a is assumed to be strictly monotone in the second variable; that is for all x ∈ Ω, for all p, q ∈ R 2 and for some constant θ > 0. Some examples of problems falling into the case are given by: (I) The equations of prescribed mean curvature: (II) The p-Laplacian: α(x, ∇u) := ∥∇u∥ p−2 , p > 1.
(III) The subsonic flow of a irrotational, ideal, compressible gas: Consider V h ⊂ H 1 0 (Ω), a class of continuous piecewise linear functions over the shape regular conforming triangulation where P 1 (T ) is the set of linear polynomial on T .Note that all T in T h are triangular elements and Ω = ∪ T ∈T h T .The Lagrange basis functions {Φ i } satisfy The discrete problem corresponding to (2) is then constructed as: find u h ∈ V h such that

A Posteriori Error Analysis
Before we get to the analysis, we would like to introduce some symbols associated with geometric information of the triangulation.We define d T the diameter of triangle T and d S the length of side S .Let S h denote the set of all interior sides of the triangulation T h .
Consider A(u; e h , v) where e h = u − u h is the error.Note that we use the abbreviations α = α(•, u) and α h = α(•, u h ) whenever convenient.By means of Green's identity, we obtain this formula Here ν T is the unit outward normal vector of T .Let the functionals R T and J S represent the element residual and the jump residual where S is the side shared by two triangles, T + and T − with the unit outward normal vectors ν + and ν − , respectively (see Figure 2), and ν S := ν − .Equation ( 7) then turns into: for all v ∈ H 1 0 (Ω).Let us define the local error indicator as and the global estimator as

Upper Bound
From the error representation (10), we obtain the upper bound for the error as follow.
Theorem 1 (Upper bound) Let u h be the approximate solution of the model problem with the error e h .Then where the constant C depends only on the shape regularity of the triangulation T h of Ω.
In order to prove Theorem 1, we need the following lemma constructed by Clément (Clément, P., 1975).
Lemma 2 (Clément's interpolations) Let T h be a shape-regular triangulation of Ω.Then there exists a linear mapping where the constant C depends only on the shape regularity of the triangulation T h , and ω T is the patch of all elements that share at least one vertex with T , see Figure 1.
Proof of Theorem 1 Let I h : H 1 0 (Ω) → V h be the L 2 -projection.If we use I h v as a test function in (7), by ( 6) we can easily show that Substitute I h v again in place of v in (10), we obtain Thus it is reasonable to rewrite (10) as Thanks to the Clément's interpolations (Lemma 2) and Cauchy-Schwarz inequality, there holds for some generic constant C > 0 depending only on regularity of the triangulation.Here ω S denotes the patch of two elements sharing the side S , see Figure 1.
Substituting e h in place of v in (17) results in The monotonic assumption (4) allows us to claim that A(u; e h , e h ) ≥ θ∥e h ∥ 2 1 , for a positive constant θ.We then finally obtain the upper estimate for the error Remark 1 Theorem 1 tells us that the error is controlled by the error estimator η h (Ω) and the oscillation from nonlinear term ∥(α h − α)∇u h ∥ 0 .Then it is helpful in designing a stopping criterion for the AFEM, if ∥(α h − α)∇u h ∥ 0 is small enough.Since α is not computable, we need some further analysis to handle with ∥(α h − α)∇u h ∥ 0 .For example, let us assume that ⃗ a is Lipschitz continuous, i.e. there is a constant c such that for all p, q ∈ V h and for any norm ∥ • ∥ defined on V h .Consider the α-approximation term: Since α is a bounded function, there is a real number M < ∞ such that If (c + M) is lower than 1, the α-approximation term in ( 19) can be ignored.

Lower Bound
In order to obtain a similar lower bound for the error, we need to estimate the indicator η h (T ) locally on T .The idea is to estimate the two components of η h (T ): ∥R T (u h )∥ 0,T and ∥J S ∥ 0,S in terms of ∥e h ∥ 1 .From now on we write R T (u h ) as R T and J S (u h ) as J S in short.With the idea of bubble functions introduced by (Verfürth, R., 1996), we obtain the following local lower bound.
Theorem 3 (Local lower bound) Let u h be the approximate solution of the model problem with the error e h .Then where ωT is the patch of elements sharing a common side with T (see Figure 1), and the oscillation on T is defined by The oscillation on a subset ω ∈ Ω is defined by Here, the constant C depends only on the shape regularity of the triangulation T h .
Before giving the proof of Theorem 3, we introduce here the notions of bubble functions used for estimations of the interior and edge residuals.
For each T ∈ T h , we define ψ T to be a polynomial function on T vanishing on ∂T and 0 ≤ ψ T ≤ 1 = max ψ T .
For each S ∈ S h , we define also χ S to be a polynomial function on ω S , as denoted in ( 17), vanishing on ∂ω S and 0 The proof of Theorem 3 relies on the properties on bubble functions as stated in the following Lemmas which are proved in (Ainsworth, M., 2000).
Lemma 4 Let P(T ) ⊂ H 1 (T ) be a finite dimensional subspace and let ψ T denote the interior bubble function over the element T .Then there exists a constant C such that for all v ∈ P(T ) where the constant C is independent of v and d T .
Lemma 5 Let P(ω S ) ⊂ H 1 (ω S ) be a finite dimensional subspace.Let S ⊂ ∂T be an edge and let χ S be the corresponding edge bubble function.Then there exists a constant C such that for all v ∈ P(ω S ) where the constant C is independent of v and d T .

Proof of Theorem 3 (a) Estimation of interior residual:
Let RT be a polynomial approximation to R T on T .Ones can see that supp(ψ T RT ) ⊂ T .It may be extended to the rest of the domain as a continuous function by defining its value outside the element to zero.The resulting extended function then belongs to H 1 0 (Ω) and thus can be used as a test function in the residual equation (10), The right hand side may be bounded with the aid of the Cauchy-Schwarz inequality.Property (24) of the interior bubble function combined with (27) help us derive for some generic constant C > 0. Applying property (23) of the interior bubble function together with the triangle inequality leads to the bound for the element residual

(b) Estimation of edge residual:
We first extend the jump residual J S , defined originally on S , constantly along the normal direction of S such that it is defined on ω S , and denote this extension by Ex(J S ) .Similarly, let JS be a polynomial approximation to the jump Ex(J S ) on the patch ω S .In the same manner, the function χ S JS having supp(χ S JS ) ⊂ ω s can be extended to the whole domain and can be used as a choice of v in (10): Now consider Each of the right hand side terms is dealt with the edge bubble function's properties and the scaled trace inequality for side S ∈ ∂T .We arrive at for some generic constant C > 0. As a consequence of (30) and the triangle inequality, we have the estimate to the jump residual The asserted estimate for η h (T ) 2 is thus obtained by the combination of the square of (30) and the sum squares of (34).
Remark 2 Ones can see from Theorem 3 that, if osc( ωT ) and ∥(α h − α)∇u h ∥ 0, ωT is sufficiently small relative to the error, the error e h will be large when the indicator η h (T ) is large.This is inline with the idea of marking strategy that mark those elements if their local indicators are higher than user's tolerance.By Remark 1, the α-approximation term can be absorbed to the error.However, the oscillation still needs to be controlled by a reasonable marking.

Adaptive Algorithm
For a given conforming shape regular triangulation T 0 along with input data α and f , the adaptive finite element algorithm proceeds as the following: Pick an initial guess u 0 with u 0 = 0 on ∂Ω, choose 0 < θ E < 1, and set h = 1.
5. Set h = h + 1 and go to Step 1.
Next, we discuss about the detail of each procedure at the h th -iteration.
The procedure SOLVE: A relatively simple way to solve nonlinear problem is the Kacanov method (Han, W.,1997) which is kind of linearlization.Given the initial approximation U 0 = u h−1 , we merely solve a sequence of linear problems for k = 1, 2, . .., instead of the nonlinear problem (6), until U K and U k−1 are close enough.Then the procedure outputs the approximate solution u h = U K .Nevertheless, some drawbacks of this fixed-point method are time consuming and requiring of some suitable assumptions for converging.
The procedure ESTIMATE: This procedure returns all quantities required in the procedure mark which are η h (T ) and osc h (T ) for all T ∈ T h .However, if necessary the approximation of the oscillation of the nonlinear term may be included.
The procedure MARK: To construct a subset Th of T h , we use the Marking Strategy E introduced by Dörfler (Dörfler, W., 1996) which guarantees the error reduction.
Marking Strategy E: Given a parameter 0 < θ E < 1, construct a minimal subset Th of T h such that ∑ and mark all elements in Th for refinement.
Another strategy used to control the oscillation reduction is the Marking Strategy O introduced by Morin et al. (Morin, P., 2000).
Marking Strategy O: Given a parameter 0 < θ O < 1 and the subset Th ⊂ T h produced by Marking Strategy E, enlarge Th to a minimal set such that ∑ and mark all elements in Th for refinement.
Remark 3 In light of the investigation in (Cascon, J. M., 2008), the rate of convergence for separate marking is suboptimal except for some range of marking parameters θ E and θ O .Observing that the indicator dominates oscillation, it is sufficient to use only the marking strategy E.
The procedure REFINE: This procedure takes the triangulation T h and the subset Th of marked elements as inputs.In order to preserve the shape regularity, all elements in Th will be refine by newest vertex bisection rule for at least n times (n ≥ 1).Of course, some more elements outside Th are also bisected to obtain a new conforming triangulation T h+1 .Notice that the resulting spaces are nested, i.e., V h ⊂ V h+1 .

Figure 1 .
Figure 2. The unit outward normal vector ν + and ν − of T + and T − on the common side S