Abstracts

Convergence Analysis of the Hessian Discretisation Method for Fourth-Order Semilinear Elliptic Equations with General Source

Devika S, BITS-Pilani, Goa Campus

In this talk, we present a convergence analysis of the Hessian Discretisation Method (HDM) for fourth-order semilinear elliptic equations involving a trilinear nonlinearity and general source terms. The HDM provides a unified framework for the convergence analysis of a broad class of numerical schemes, including conforming and nonconforming finite element methods as well as gradient recovery based methods.

We establish convergence results using two complementary approaches. The analysis is based on four key properties of the HDM, together with a suitable companion operator. Notably, the first approach does not require additional regularity assumptions on the exact solution or the assumption that the solution is regular, whereas the second approach, under the above assumptions, yields explicit orders of convergence.

We further illustrate the applicability of the theoretical results through two important examples: the Navier–Stokes equations in the stream function–vorticity formulation and the von Kármán equations for plate bending. Finally, numerical results are presented to support the theoretical estimates.