This work aims to develop a fast and spatially fourth-order Cartesian grid finite difference method for solving elliptic and parabolic problems over two-dimensional irregular domains with sharply curved boundaries, under the assumptions that the boundary is \(C^1\) -continuous and the solution is sufficiently smooth up to the boundary. The proposed Augmented Matched Interface and Boundary (AMIB) method inherits its predecessor’s speed and accuracy advantages, such as maintaining the Fast Fourier-Transform (FFT) efficiency and being fourth-order accuracy in handling any boundary conditions (Dirichlet, Neumann, Robin, or their combinations). To accommodate sharply curved boundaries, the proposed adaptive AMIB method features two significant improvements. First, an adaptive ray-casting Matched Interface and Boundary (MIB) scheme was developed to overcome the difficulties of generating fictitious values at some grid points where the boundary is sharply curved by reusing previously calculated fictitious values at nearby grid points. Second, several stabilizers, which include a new preconditioner for the resulting augmented system and proper grid selection requirements to interpolate the fictitious values and approximate derivative jumps in the MIB scheme, were designed to ensure the stability of the AMIB method. Numerical experiments have been conducted to validate the proposed AMIB method for solving boundary and initial value problems with sharply curved boundaries.
