A stability-enhanced preconditioned Haar wavelet scheme for the solution of Poisson and Helmholtz equations

This study presents a modified numerical framework based on scale-3 Haar wavelets for the solution of elliptic partial differential equations, namely the Poisson and the Helmholtz equations. The spatial derivatives are discretized using Haar wavelet expansions and extended to two-dimensions through the Kronecker tensor product, with boundary conditions enforced via integration constants. A theoretical convergence analysis is conducted, and complemented by a rigorous stability investigation based on both classical and effective condition numbers to address ill-conditioning in the resulting linear systems at higher resolution levels. For the Poisson equation, the effectiveness of wavelet-based and multigrid preconditioners is examined, which shows that the multigrid preconditioner yields superior performance compared to the Haar wavelet preconditioner for scale-2 Haar discretization, while the Haar wavelet preconditioner performs better for scale-3 discretization when combined with the conjugate gradient method. For the Helmholtz equation, a shifted Laplacian preconditioner coupled with the GMRES solver is employed. Numerical simulations and spectral analyses, implemented in MATLAB R2025a, confirm the enhanced stability and convergence of the proposed preconditioned wavelet framework.