On the finite difference method with penalty for numerical solution of PDEs over curved domains

This paper aims to decipher the elliptic PDEs defined over curved complex domains with a penalty approach in a finite difference scheme. Using the penalty approach, we utilize the finite difference scheme with the aid of uniform Cartesian mesh irrespective of the shape and convexity of the domain. We introduce the finite difference schemes with the \(H^1\) $H^1$ and \(L^2\) $L^2$ penalties and furnish the sharp convergence of the solution of the penalized problems to the solution of the original problem. Several iterative solvers are also equipped for both the \(H^1\) $H^1$ and \(L^2\) $L^2$ penalty schemes to compare the required number of iterations and CPU time to achieve the desired accuracy. The comprehensive numerical experiments are performed to uphold the mathematical results and to exhibit the efficiency of the proposed idea for practical scenarios.