Volume 32, Issue 1

Here, we extend the finite distortion problem from bounded domains in ℝ 2 to closed genus-zero surfaces in ℝ 3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M $\mathcal{M}$ and a unit sphere 𝕊 2 by minimizing the total area distortion energy on ℂ . After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M $\mathcal{M}$ and 𝕊 2 . The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M $\mathcal{M}$ and 𝕊 2 . In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.

This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux ( F * ) and a non-oscillatory flux ( F s ). This novel approach paves a way to construct non-oscillatory entropy stable flux (Fˆ) $(\widehat{F})$as a simple combination of ( F * and F s ) which inherently optimize the numerical diffusion in the entropy stable flux (Fˆ) $(\widehat{F})$such that it reduces to the underlying non-oscillatory flux ( F s ) in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair ( F * , F s ), (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that fully discrete schemes using these entropy stable fluxes do not exhibit nonphysical spurious oscillations in approximating the discontinuities and its non-oscillatory nature is comparable to the non-oscillatory schemes using underlying fluxes ( F s ) only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair.

We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n -tensor heat flow problem. These methods differ with respect to the surface representation used, the geometric information required, and the treatment of the tangentiality condition. We emphasize the importance of geometric properties and their increasing influence as the tensorial degree changes from n = 0 to n ⩾ 1. A specific example is presented that illustrates how curvature drastically affects the behavior of the solution.

An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residual-based a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C 0 interior penalty, as well as weakly over-penalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H −2 (Ω).