scholarly journals A Posteriori Estimates for Errors of Functionals on Finite Volume Approximations to Solutions of Elliptic Boundary-Value Problems

2013 ◽  
pp. 57-68
Author(s):  
Lutz Angermann
Keyword(s):  
Boundary Value Problems ◽  
Finite Volume ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
A Posteriori ◽  
Elliptic Boundary Value ◽  
A Posteriori Estimates

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.

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