## Error estimates on anisotropic Q1 elements for functions in weighted Sobolev spaces

Venue: | Math. Comp |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Durán_errorestimates,

author = {Ricardo G. Durán and Ariel and L. Lombardi},

title = {Error estimates on anisotropic Q1 elements for functions in weighted Sobolev spaces},

journal = {Math. Comp},

year = {},

pages = {1679--1706}

}

### OpenURL

### Abstract

Abstract. InthispaperweproveerrorestimatesforapiecewiseQ1average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions. Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses. Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems. Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements. As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained. 1.

### Citations

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Citation Context ...ctangular elements, i.e., rectangles with sides of different orders. The classic error analysis is based on the so-called regularity assumption which excludes these kinds of elements (see for example =-=[8, 9]-=-). However, it is now well known that this assumption is not needed. Indeed, many papers have been written to prove error estimates under more general conditions. In particular, for rectangular elemen... |

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Citation Context ...ctangular elements, i.e., rectangles with sides of different orders. The classic error analysis is based on the so-called regularity assumption which excludes these kinds of elements (see for example =-=[8, 9]-=-). However, it is now well known that this assumption is not needed. Indeed, many papers have been written to prove error estimates under more general conditions. In particular, for rectangular elemen... |

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Citation Context ...ite element approximations of singular problems is a well-known procedure. In particular, error estimates for functions in weighted Sobolev spaces have been obtained in several works (see for example =-=[2, 5, 6, 14]-=-). In those works, the weights considered are related to the distance to a point or an edge (in the 3D case); instead here we consider weights related to the distance to the boundary. Finally, we cons... |

40 | The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges
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(Show Context)
Citation Context ...ite element approximations of singular problems is a well-known procedure. In particular, error estimates for functions in weighted Sobolev spaces have been obtained in several works (see for example =-=[2, 5, 6, 14]-=-). In those works, the weights considered are related to the distance to a point or an edge (in the 3D case); instead here we consider weights related to the distance to the boundary. Finally, we cons... |

21 |
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Citation Context ...‖L2 (0,1) for v ∈ H1 0 (0, 1). We will also need the following generalization to higher dimensions: If D is a convex domain and u ∈ H1 0 (D), then (2.4) ∥ u ∥ ≤ 2‖∇u‖L2 (D) L2 (D) dD (see for example =-=[17]-=-). The following lemma gives an “anisotropic” version of (2.4). It can be proved by standard scaling arguments. Lemma 2.1. Let R =Πd i=1 (ai,bi) be a d-rectangle and hi = bi − ai, 1 ≤ i ≤ d. For all u... |

19 | Anisotropic mesh refinement for singularly perturbed reaction diffusion problems
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(Show Context)
Citation Context ...imates follow from our results of Sections 2 and 3. The meshes that we construct are very different from the Shishkin type meshes that have been used in other papers for this problem (see for example =-=[4, 16]-=-). In particular, our almost optimal error estimate in the energy norm is obtained with meshes independent of ε.1698 R. G. DURÁN AND A. L. LOMBARDI Given a partition Th of (0, 2) × (0, 2) into rectan... |

13 | Interpolation of non-smooth functions on anisotropic finite element meshes
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Citation Context ...the estimates ∂ ∣ ku ∂xk { (x1,x2) ∣ ≤ C 1+ε 1 −k e − x1 ε + ε −k e − 2−x } 1 (4.29) ε , ∂ ∣ k u { (x1,x2) ∣ ≤ C 1+ε −k e − x2 ε + ε −k e − 2−x } 2 (4.30) ε ∂x k 2 provided that 0 ≤ k ≤ 4and(x1,x2) ∈ =-=[0, 2]-=- × [0, 2], which are proved in [16]. As an example we prove the first inequality in (4.5). Observe that, for r =0, 1, 2, (x1,x2) ≡ 0whenx2 =0orx2 =2fori=1andwhenx1=0orx1 =2for ∂ r u ∂x r i □ERROR EST... |

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Citation Context ...:= u − ( ∫ R u)ψ has vanishing mean value, there exists F ∈ H1 0 (R)d such that (2.7) −div F = v and (2.8) ‖F ‖ H 1 0 (R) 2 ≤ C‖v‖ L 2 (R). Moreover, from the explicit bound for the constant given in =-=[13]-=- it follows that C can be taken depending only on δ. Now, since ∫ uψ = 0, we have from (2.7) R ‖u‖ 2 L2 (R) = ∫ ∫ uv = − u div F R and therefore, integrating by parts and using (2.4) for each componen... |

10 |
Uniform error estimates for certain narrow Lagrange finite elements
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Citation Context ...is now well known that this assumption is not needed. Indeed, many papers have been written to prove error estimates under more general conditions. In particular, for rectangular elements we refer to =-=[1, 12, 18]-=- and their references. Received by the editor August 4, 2003 and, in revised form, January 8, 2004. 2000 Mathematics Subject Classification. Primary 65N30. Key words and phrases. Anisotropic elements,... |

8 | and M.F.Wheeler, Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids
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(Show Context)
Citation Context ...ngular perturbation model problem (4.1) −ε 2 ∆u + u = f in (0, 2) × (0, 2), u =0 on ∂{(0, 2) × (0, 2)}. Compatibility conditions are assumed in order to have the regularity results proved in [15] and =-=[16]-=-. As we will show, appropriate graded anisotropic meshes can be defined in order to obtain almost optimal order error estimates in the energy norm valid uniformly in the parameter ε. These estimates f... |

4 |
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(Show Context)
Citation Context ...olations satisfying Dirichlet boundary conditions to other domains is not straightforward and would require further analysis. To prove the weighted estimates, we will use a result of Boas and Straube =-=[7]-=- which, as we show, can be derived from the classic Hardy inequality in higher dimensions. In Section 2 we construct the mean average interpolation and prove the error estimates for interior elements.... |

4 |
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(Show Context)
Citation Context ...is now well known that this assumption is not needed. Indeed, many papers have been written to prove error estimates under more general conditions. In particular, for rectangular elements we refer to =-=[1, 12, 18]-=- and their references. Received by the editor August 4, 2003 and, in revised form, January 8, 2004. 2000 Mathematics Subject Classification. Primary 65N30. Key words and phrases. Anisotropic elements,... |

2 |
and average interpolation over 3D anisotropic elements
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(Show Context)
Citation Context ...is now well known that this assumption is not needed. Indeed, many papers have been written to prove error estimates under more general conditions. In particular, for rectangular elements we refer to =-=[1, 12, 18]-=- and their references. Received by the editor August 4, 2003 and, in revised form, January 8, 2004. 2000 Mathematics Subject Classification. Primary 65N30. Key words and phrases. Anisotropic elements,... |

1 |
Differentiability properties of solutions of the equation −ɛ2∆u + ru = f(x, y) in a square, SIAMJ.Math.Anal.21
- Han, Kellogg
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(Show Context)
Citation Context ...on the singular perturbation model problem (4.1) −ε 2 ∆u + u = f in (0, 2) × (0, 2), u =0 on ∂{(0, 2) × (0, 2)}. Compatibility conditions are assumed in order to have the regularity results proved in =-=[15]-=- and [16]. As we will show, appropriate graded anisotropic meshes can be defined in order to obtain almost optimal order error estimates in the energy norm valid uniformly in the parameter ε. These es... |