A reconstruction based exponentially accurate spectral method to solve Elliptic Partial Differential Equations with discontinuous coefficients and complex shaped interfaces

Elliptic Partial Differential Equations (EPDE) with discontinuous coefficients are essential for modelling physical phenomena in media with varying material properties across sharp interfaces with complex geometry. The well-known exponential convergence of spectral methods for differential equations, reduces to algebraic convergence due to the Gibbs-Wilbraham phenomenon due to the discontinuity, thus limiting their application to EPDEs. To overcome Gibbs-Wilbraham phenomenon we propose a reconstruction technique that decomposes the solution into a C^∞ smooth function and a modified Heaviside function, where a C^∞ correction function modifies the Heaviside step function. In our previous work we proposed a weak formulation for the correction function. We now propose a novel strong form approach where the correction function is obtained by solving the Cauchy problem spanning over sub-domains separated by the interface and imposing conditions on the interface. The smooth function, is also obtained from the solution of the EPDE without the interface. We showcase exponential accuracy while resolving discontinuities at a sharp interfaces using the Helmholtz and Poisson equations for two- and three-dimensional problems, from acoustics and fluid dynamics.