scholarly journals A least-squares Galerkin approach to gradient and Hessian recovery for nondivergence-form elliptic equations

2021 ◽  
Author(s):  
Omar Lakkis ◽  
Amireh Mousavi
Keyword(s):  
Least Squares ◽  
Elliptic Equations ◽  
Adaptive Mesh Refinement ◽  
Least Squares Method ◽  
A Priori ◽  
Adaptive Mesh ◽  
A Posteriori ◽  
Nondivergence Form ◽  
Convergence Results ◽  
Discrete Spaces

Abstract We propose a least-squares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax–Milgram theorem with the curl-free constraint built into the target (or cost) functional, the discrete spaces require no inf-sup stabilization. We show that standard conforming finite elements can be used yielding a priori and a posteriori convergence results. We illustrate our findings with numerical experiments with uniform or adaptive mesh refinement.

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

2004 ◽  
Vol 4 (1) ◽  
pp. 105-127 ◽  
Author(s):  
Grigory I. Shishkin ◽  
Lidia P. Shishkina ◽  
Pieter W. Hemker
Keyword(s):  
Coordinate System ◽  
Transition Layer ◽  
A Priori ◽  
Adaptive Mesh ◽  
Singularly Perturbed ◽  
Finite Difference Schemes ◽  
A Posteriori ◽  
Interior Layer ◽  
Convection Diffusion ◽  
Mesh Technique

AbstractWe study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’.

scholarly journals Residual-based a posteriori error analysis for the coupling of the Navier-Stokes and Darcy-Forchheimer equations

2021 ◽  
Author(s):  
Sergio Caucao ◽  
Gabriel Gatica ◽  
Ricardo Oyarzúa ◽  
Felipe Sandoval
Keyword(s):  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Posteriori Error ◽  
Local Approximation ◽  
Navier Stokes ◽  
Error Estimator ◽  
A Posteriori ◽  
Approximation Properties ◽  
A Posteriori Error ◽  
Posteriori Error Analysis

In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier--Stokes and Darcy--Forchheimer equations, and derive,  though in a non-standard sense,  a reliable and efficient residual-based a posteriori error estimator suitable for an adaptive mesh-refinement method.  For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Cl\'ement interpolant and Raviart--Thomas operator, and a smallness assumption on the data.   In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in local $\L^\rp$ spaces, are the main tools for developing the effi\-ciency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

Some Open Questions in the Numerical Analysis of Singularly Perturbed Differential Equations

2015 ◽  
Vol 15 (4) ◽  
pp. 531-550 ◽  
Author(s):  
Hans-Görg Roos ◽  
Martin Stynes
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Singularly Perturbed ◽  
Defect Correction ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Open Questions ◽  
Convergence Results

AbstractSeveral open questions in the numerical analysis of singularly perturbed differential equations are discussed. These include whether certain convergence results in various norms are optimal, when supercloseness is obtained in finite element solutions, the validity of defect correction in finite difference approximations, and desirable adaptive mesh refinement results that remain to be proved or disproved.

FE Meshing Tool for the Dynamic Analysis of Beam-Like Structures

1991 ◽  
Author(s):  
M. De Smet ◽  
H. Van Brussel ◽  
P. Sas
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Adaptive Mesh Refinement ◽  
Dynamic Behaviour ◽  
A Priori ◽  
Adaptive Mesh ◽  
Structural Dynamic ◽  
Resonant Frequencies ◽  
Tapered Beam ◽  
Mesh Density

Abstract This paper outlines a methodology for the automatic creation of finite element meshes used for calculating structural dynamic behaviour. Starting from the idealized geometry of the structure, the proposed Intelligent Meshing Tool determines a mesh density that a priori guarantees the resonant frequencies to be calculated within predetermined accuracy limits. Unlike most existing techniques for automatic meshing, this technique is not inherently adaptive, but does not exclude an extra adaptive mesh refinement step, either. Implementation of this methodology is discussed for structures composed of straight uniform, curved and tapered beam-like components. The feasibility of the meshing strategy is illustrated by means of an example.

scholarly journals An Optimum Deconvolution Method

1979 ◽  
Vol 49 ◽  
pp. 287-290
Author(s):  
C.R. Subrahmanya
Keyword(s):  
Least Squares ◽  
Prior Knowledge ◽  
Least Squares Method ◽  
A Priori ◽  
Statistical Treatment ◽  
Stable Solution ◽  
Brightness Distribution ◽  
Deconvolution Method ◽  
Explicit Recognition ◽  
Optimum Solution

An optimum solution to a deconvolution problem has to fulfil three general criteria: (a) an explicit recognition of the smoothing nature of convolution; (b) a statistical treatment of noise, e.g., using the least-squares criterion; and (c) requiring the solution to conform to all our prior knowledge about it. In the usual least-squares method, one minimises a variance of ‘residuals’, or the departures of the observed data from the values expected according to the recovered solution. However, this condition does not lead to a stable solution in the case of deconvolution, since the only stable solutions are those conforming to a criterion of ‘regularisation’ or smoothness (see, e.g., Tikhonov and Arsenin 1977). In our method, the stability is achieved by minimising the variance of the second-differences of the solution simultaneously with the fulfilment of the least-squares criterion. Such a procedure was first used by Phillips(1962). However, the solution thus obtained is still unsatisfactory since it usually does not conform to oura prioriinformation. When we seek the brightness distribution of an object, the most frequent violation of our prior knowledge is that of positiveness. This motivated us to develop an Optimum Deconvolution Method (ODM) which constrains the solution to satisfy prior knowledge while retaining the features of least-squares and smoothness criteria.

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