arXiv subjects

Aamir Yousuf

Publications and source records attributed to Aamir Yousuf.

4 recordsLinked to original sources

arXiv2026

Unconditionally stable and energy conserving discretization of the dynamic von Kármán equations

A fully discrete approximation of the dynamic von Kármán equations combines nonconforming Morley finite element methods for spatial discretization with an energy conserving modified unconditionally stable Newmark second- order time-stepping scheme. Brouwer's fixed-point theorem establishes existence of a solution to the fully discrete scheme and further uniqueness and stability estimates follow for small loads. Optimal order a priori error estimates in the piecewise energy norm with quadratic convergence in time are derived for the fully discrete scheme. The results of the numerical experiments validate the theoretical error bounds.

math.NA↗

arXiv2026

Unified numerical analysis for thermoelastic diffusion and thermo-poroelasticity of thin plates

We investigate a coupled hyperbolic-parabolic system modeling thermoelastic diffusion (resp. thermo-poroelasticity) in plates, consisting of a fourth-order hyperbolic partial differential equation for plate deflection and two second-order parabolic partial differential equations for the first moments of temperature and chemical potential (resp. pore pressure). The unique solvability of the system is established via Galerkin approach, and the additional regularity of the solution is obtained under appropriately strengthened data. For numerical approximation, we employ the Newmark method for time discretization of the hyperbolic term and a continuous interior penalty scheme for the spatial discretization of displacement. For the parabolic equations that represent the first moments of temperature and chemical potential (resp. pore pressure), we use the Crank--Nicolson method for time discretization and conforming finite elements for spatial discretization. The convergence of the fully discrete scheme with quasi-optimal rates in space and time is established. The numerical experiments demonstrate the effectiveness of the 2D Kirchhoff--Love plate model in capturing thermoelastic diffusion and thermo-poroelastic behavior in specific materials. We illustrate that as plate thickness decreases, the two-dimensional simulations closely approximate the results of three-dimensional problem. Finally, the numerical experiments also validate the theoretical rates of convergence.

math.NA↗

arXiv2026

Hybrid-high order method in space and implicit schemes in time for the biharmonic wave equation

This article presents the numerical analysis for the biharmonic wave equation with clamped boundary conditions employing two variants of the {hybrid high-order} method for the space discretization and two implicit time-stepping schemes for the time discretization. The Newmark scheme directly discretizes the second-order time derivative, while the Crank-Nicolson scheme discretizes a reformulated system where we introduce velocity as an independent variable to create coupled first-order equations. Optimal orders of convergence in space and time are achieved for both schemes. The numerical experiments validate the theoretical convergence rates and show the effectiveness of the proposed methods. To the best of our knowledge, this is the first work in literature that addresses hybrid-high order method and implicit time schemes for the biharmonic wave equation.

math.NA↗

arXiv2025

Semi and fully-discrete analysis of lowest-order nonstandard finite element methods for the biharmonic wave problem

This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz projection operator defined on $H^2_0(Ω)$ ensures error estimates under appropriate regularity assumptions on the solution. Stability results and error estimates of optimal order are established in suitable norms for the semidiscrete and explicit/implicit fully-discrete versions of the proposed schemes. Finally, we report on numerical experiments using explicit and implicit schemes for time discretization and Morley, discontinuous Galerkin, and {C$^0$ interior} penalty schemes for space discretization, that validate the theoretical error estimates.

math.NA↗