On the Finite Element Method with Riesz Bases and Its Applications to Some Partial Differential Equations

In this paper, we will find the numerical solution of partial differential equations by using the finite element method with Riesz bases that are elevated from ortho normal bases. Especially for the two-dimensional cases, we also propose another way to solve a matrix-valued equation (Lyapunov equation). Moreover, we give the precise inverse formula for symmetric block tridiagonal Toeplitz matrices. To conclude, we present some numerical results that show the usefulness of the elevated bases.