A Posteriori Error Estimation for Hierarchic Models of Elliptic Boundary Value Problems on Thin Domains

1996 ◽  
Vol 33 (1) ◽  
pp. 221-246 ◽  
Author(s):  
I. Babuška ◽  
C. Schwab
Keyword(s):  
Boundary Value Problems ◽  
Error Estimation ◽  
Elliptic Boundary ◽  
Posteriori Error ◽  
Elliptic Boundary Value Problems ◽  
A Posteriori ◽  
Posteriori Error Estimation ◽  
Thin Domains ◽  
A Posteriori Error ◽  
Elliptic Boundary Value

scholarly journals A posteriori verification of the positivity of solutions to elliptic boundary value problems

2022 ◽  
Vol 3 (1) ◽  
Author(s):  
Kazuaki Tanaka ◽  
Taisei Asai
Keyword(s):  
Boundary Value Problems ◽  
Error Estimation ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
A Priori ◽  
Previous Method ◽  
Elliptic Boundary Value Problems ◽  
A Posteriori ◽  
Elliptic Boundary Value ◽  
Positivity Of Solutions

AbstractThe purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $$H^2$$ H 2 -regularity nor $$ L^{\infty } $$ L ∞ -error estimation, but only $$ H^1_0 $$ H 0 1 -error estimation. In (J Comput Appl Math 370:112647, 2020), we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require $$ L^{\infty } $$ L ∞ -error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.

scholarly journals ENERGY NORM A POSTERIORI ERROR ESTIMATION OF hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS

2007 ◽  
Vol 17 (01) ◽  
pp. 33-62 ◽  
Author(s):  
PAUL HOUSTON ◽  
DOMINIK SCHÖTZAU ◽  
THOMAS P. WIHLER
Keyword(s):  
Discontinuous Galerkin ◽  
Error Estimation ◽  
Elliptic Boundary ◽  
Posteriori Error ◽  
A Posteriori Error Estimation ◽  
Energy Norm ◽  
Upper And Lower Bounds ◽  
A Posteriori ◽  
Posteriori Error Estimation ◽  
A Posteriori Error

In this paper, we develop the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hp-adaptive refinement procedure.

scholarly journals ENERGY NORM A POSTERIORI ERROR ESTIMATION FOR hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS IN THREE DIMENSIONS

2011 ◽  
Vol 21 (02) ◽  
pp. 267-306 ◽  
Author(s):  
LIANG ZHU ◽  
STEFANO GIANI ◽  
PAUL HOUSTON ◽  
DOMINIK SCHÖTZAU
Keyword(s):  
Discontinuous Galerkin ◽  
Error Estimation ◽  
Elliptic Boundary ◽  
Posteriori Error ◽  
A Posteriori Error Estimation ◽  
Three Dimensions ◽  
Energy Norm ◽  
A Posteriori ◽  
Posteriori Error Estimation ◽  
A Posteriori Error

We develop the energy norm a posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the error measured in terms of the natural energy norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of an hp-version averaging operator in three dimensions, which we explicitly construct and analyze. We use our error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.

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