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Adaptive Finite Element Methods for Elliptic Partial Differential Equations

Sofia Petrova, Dmitri Volkov

In: Recent Advances in Numerical Methods and Computational Mathematics

We present a novel a posteriori error estimation framework for adaptive finite element methods applied to second-order elliptic PDEs. Our approach combines residual-based error indicators with hierarchical basis functions, achieving optimal convergence rates on both structured and unstructured meshes. Numerical experiments on benchmark problems confirm the efficiency and robustness of the proposed algorithm.

Stochastic Differential Equations in Financial Mathematics

Amara Patel, Sunita Krishnan

In: Mathematical Modeling in Physical Sciences

This paper reviews stochastic differential equations (SDEs) and their applications in financial modeling. We examine the Black-Scholes framework, Heston stochastic volatility model, and jump-diffusion processes. Numerical methods for SDE simulation including Euler-Maruyama, Milstein, and Runge-Kutta schemes are analyzed with convergence guarantees.