Matrix Science Mathematic (MSMK) 3(2) (2019) 27-31 Cite The Article: Y. Kashwaa, A. Elsaid, M. El-Agamy (2019) Weighted Technique For Finite Element Gradient Recovery At Boundary. Matrix Science Mathematic, 3(2): 27-31. ARTICLE DETAILS Article History: Received 27 December 2019 Accepted 28 January 2020 Available online 07 February 2020 ABSTRACT In this paper, an improved technique is presented to recover the finite element gradient at boundaries. The proposed technique begins by evaluating the recovered gradient at the interior nodes using polynomial preserving recovery technique. Then we propose formula for weights to the recovered gradient at the interior nodes attached to boundary nodes. The sum of these weighted recovered gradients is utilized as an approximation for the gradient at the attached boundary node. The validity of the proposed technique is illustrated by some two-dimensional numerical examples. KEYWORDS Gradient at boundary, Polynomial preserving recovery, Finite element method, Poisson equation. 1. INTRODUCTION Gradient recovery techniques are post-processing methods that reconstruct numedical approximations from finite element solution to obtain a better numerical gradient (recovered gradient). Several ideas for gradient recovery techniques were proposed by researchers as the work in [1-3]. But these techniques suffered some drawbacks which include long execution time, complexity, requiring meshes with special structure, or the necessity of using low-order finite elements. The first technique to overcome the above-mentioned drawbacks was developed in [4] and called the super convergent patch recovery (SPR). The advantages that are noticed in SPR include super convergence of the recovered gradients, its robustness as a posteriori error estimator, and its efficient execution [5- 11]. In some studies the polynomial preserving recovery (PPR) technique was proposed and it was demonstrated that the PPR is better than the SPR [7,11]. The idea behind recovering the gradient using the PPR technique starts by construction a patch of elements around the targeted node. Then, solution at the nodes of this patch of elements is fitted to a second-order polynomial using least-squares approach. The gradient of this fitting polynomial is the recovered gradient at this node. The results obtained using PPR often yields a higher-order gradient approximation on the patch of mesh elements around each mesh vertex. On the other hand, the accuracy of PPR near boundaries of the considered domain is not as good as that obtained at nodes in the interior of the domain. So, some treatments are required to improve the accuracy of PPR on boundaries. A gradient recovery technique on boundary node was proposed [13]. The authors utilized the PPR least squares fitting technique to construct a polynomial for each selected interior node attached to the target boundary node. Then they computed the recovered gradient at this boundary node as the gradient of the polynomial obtained by averaging the polynomials constructed at the attached internal nodes. The numerical examples they presented show that in some cases the recovered gradient evaluated using this technique has better accuracy than the classical PPR. In this paper, we propose a weighted technique to obtain a better recovered gradient at nodes on the boundary. Our technique is performed in two stages: First the standard PPR is applied to each selected interior node attached to the target boundary node. Then we utilize an error estimator to assign weights for the obtained recovered gradients and substitute with the boundary node in the sum of these weighted recovered gradients. The rest of this article is organized as follows. In section 2, the PPR method and the procedure of adaptive finite element method are discussed. In section 3, we introduce the mathematical formulation of our technique. Section 4 contains some numerical examples for applying the proposed technique to Poisson equation in two-dimensional domains. Polygonal domains with different types of corners are considered and results are shown for both uniform and adaptive refinement. Finally section 5 presents the conclusion of this work. 2. MATHEMATICAL PRELIMINARIES 2.1 Basic concepts of PPR Let v denote a mesh node, n denote a positive integer, and ℓ(v, n) denote the union of elements in the first n layers around v, i.e., ℓ(v, n) =∪: {τ ∈ τh, τ ∩ ℓ(v, n−1 ) = ∅}, (1) where ℓ (v,0) = {v} and τh denotes consequence of triangulation of the finite element mesh. Let the space of continuous linear finite element space and the set of all mesh nodes be denoted by Vh and Nh, respectively. The standard Lagrange basis of Vh is denoted by {φv,v∈Nh} with {φv(v') = δvv'}. Let the gradient recovery PPR operator be denoted by Gh such that Gh: Vh → Vh × Vh. For a mesh node v, let αv be a patch of elements around v. The polynomial   ∈  +1 (  ) is to be fitted in the least square sense at sampling points that consist of the set of all nodes in  ℎ ∩   , i.e.   = arg min   ∈ +1 (  ) ∑ ( ℎ − ) 2 (̅ ), (2)   ∈ ℎ ∩   where   (  ) is the space of polynomials defined on αv with a degree less than or equal to k. Then (Ghuh)(v) = ∇Pv(v). (3) The finite element representation of the recovered gradient on the entire domain is given by ( ℎ  ℎ ) = ∑ ( ℎ  ℎ )() ∈ ℎ   . (4) For an interior node v, we define ϑv as the smallest ℓ(v, n) that ensures the uniqueness of the fitting polynomial pv [6,7]. In the case that v is a boundary node, let n0 be the smallest positive integer such that ℓ(v, n0) has at least one interior mesh node. Then, we define Matrix Science Mathematic (MSMK) DOI : http://doi.org/10.26480/msmk.02.2019.27.31 WEIGHTED TECHNIQUE FOR FINITE ELEMENT GRADIENT RECOVERY AT BOUNDARY Y. Kashwaa, A. Elsaid*, M. El-Agamy Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University, P.O. 35516, Mansoura, Egypt. *Corresponding Author E-mail: a_elsaid@mans.edu.eg This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ISSN: 2521-0831(Print) ISSN: 2521-084X (Online) CODEN : MSMADH RESEARCH ARTICLE