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

The 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.

Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum

This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

An isogeometric mortar method for the coupling of multiple NURBS domains with optimal convergence rates

We investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patches $$\Omega _k$$ Ω k with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces $$X_{h,k}$$ X h , k on each patch $$\Omega _k$$ Ω k as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an extension of Brivadis et al. (Comput Methods Appl Mech Eng 284:292–319, 2015) and highlights several aspects which did not receive full attention before. In particular, by choosing appropriate spaces of polynomial splines as Lagrange multipliers, we obtain a uniform infsup-inequality. Moreover, we provide a new additional condition on the discrete spaces $$X_{h,k}$$ X h , k which is required for obtaining optimal convergence rates of the mortar method. Our numerical examples demonstrate that the optimal rate is lost if this condition is neglected.

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify on one hand the regularity of the solution with respect to the random domain mapping required for many multilevel quadrature methods, such as the multilevel quasi-Monte Carlo quadrature using Halton points, the multilevel sparse anisotropic Gauss–Legendre and Clenshaw–Curtis quadratures and multilevel interlaced polynomial lattice rules. On the other hand, we derive the coupling formulation and show by numerical results that the approach is feasible.