A posteriori verification of the positivity of solutions to elliptic problems
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The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions u of elliptic problems while assuming H^1_0-error estimation u-û_H_0^1≤ρ given some numerical approximation û and an explicit error bound ρ. In [J. Comput. Appl. Math, Vol. 370, (2020) 112647], we achieved a method of verifying positivity for several semilinear elliptic problems by unifying two approaches. However, some cases also require L^∞-error estimation u-û_L^∞≤σ given explicit σ>0. Therefore, the range of application is somewhat limited. 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 calculated from approximation û because the proposed method requires explicit estimates of them. We also numerically apply our method to several problems, including those to which the previous method is not applicable.
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