  We develop a universally applicable embedded boundary finite difference
method, which results in a symmetric positive definite linear system and does
not suffer from small cell stiffness. Our discretization is efficient for the
wave, heat and Poisson's equation with Dirichlet boundary conditions. When the
system needs to be inverted we can use the conjugate gradient method,
accelerated by algebraic multigrid techniques. A series of numerical tests for
the wave, heat and Poisson's equation and applications to shape optimization
problems verify the accuracy, stability, and efficiency of our method. Our fast
computational techniques can be extended to moving boundary problems (e.g.
Stefan problem), to the Navier-Stokes equations, and to the Grad-Shafranov
equations for which problems are posed on domains with complex geometry and
fast simulations are very important.
