Explicit Constants in Poincaré-Type Inequalities for Simplicial Domains and Application to A Posteriori Estimates

Abstract

The paper is concerned with sharp estimates of constants in the classical Poincaré inequalities and Poincaré-type inequalities for functions with zero mean values in a simplicial domain or on a part of the boundary. These estimates are important for quantitative analysis of problems generated by differential equations where numerical approximations are typically constructed with the help of simplicial meshes. We suggest easily computable relations that provide sharp bounds of the respective constants and compare these results with analytical estimates (if such estimates are known). In the last section, we discuss possible applications and derive a computable majorant of the difference between the exact solution of a boundary value problem and an arbitrary finite dimensional approximation defined on a simplicial mesh.

Spectral gap estimates for Brownian motion on domains with sticky-reflecting boundary diffusion

“…where C Ω and C ∂Ω are the usual (Neumann) Poincaré constants of Ω and ∂Ω respectively. To derive this result we combine Escobar's lower bound [9] on the first Steklov eigenvalue [12,20] of Ω with a novel estimate on the optimal zero mean trace Poincaré constant of Ω [22,26], for which we obtain that…”

Estimates of the Distance to Exact Solutions of the Stokes Problem with Slip and Leak Boundary Conditions

“…For such domains as parallelepipeds, triangles, and tetrahedrons, the constant C Γ 2 has been found analytically or numerically (cf. [36,37] and the references therein). It is worth noting that the constants C Γ 2 (Ω) possess a certain monotonicity property, namely, if Ω ⊂ Ω and both domains have the same boundary part Γ 2 (where the functions have traces with zero mean), then C Γ 2 (Ω) C Γ 2 (Ω ).…”

“…Using these results and applying affine mappings, one can deduce the constants for any other nondegenerate triangle (cf. [36]). Therefore, finding the constants is not a difficult task (at least in the case where Γ 2 is a boundary of a polygonal domain).…”

Thin obstacle problem: Estimates of the distance to the exact solution

“…The constants C P M (Ω ± ) entering m ± (q * ) are also easy to estimate. These constants are known for triangles (see [NR15] and [MR16]). Due to this fact, we can easily obtain upper bounds of the constants for a wide collection of domains.…”