galerkin method with modified shifted lucas polynomials for solving the 2d poisson equation

This study looks at how to solve the two-dimensional poisson equation, a math problem common in physics and engineering. we focus on spectral methods, which are good at solving problems with smooth solutions. we introduce a spectral galerkin method that uses tensor products of modified shifted lucas polynomials. these polynomials haven’t been used this way before. by adding a factor of x(1-x) to the lucas polynomials, our method automatically meets certain boundary conditions, which makes it easier to use while keeping its accuracy. our goal is to create and test this method for solving the poisson equation on a square. we create fast algorithms for putting together matrices and study how well the method converges using math and computer experiments. the tests show that our method has similar convergence rates to other methods like chebyshev and legendre. the errors go down exponentially for smooth source terms. the method is efficient and has good conditioning, which suggests that lucas polynomials could be a good alternative to regular polynomials in spectral methods. this research could lead to using lucas polynomial-based spectral methods for more general problems.