Spectral element methods for parabolic problems

Nonconforming Least-Squares Spectral Element Method for European Options

“…Schötzau et al [8] proposed hp-version of the Discontinuous Galerkin Finite Element Method to solve parabolic problems. In [9], Dutt et al proposed Least-Squares Spectral Element Method for parabolic partial differential equations (PDE) on bounded domains and proved exponential accuracy for analytic data.…”

Preconditioners for spectral element methods for elliptic and parabolic problems

“…Moreover in [2] it has been shown that a preconditioner can be defined for the quadratic form corresponding to the minimization problem which allows the problem to decouple. Let u( , , ) be a polynomial of degree N in each of the space variables and and of degree M in the time variable separately.…”

“…The method is a least-squares method as formulated in [2][3][4]9,10]. We minimize at each time step a functional which is the sum of the squares of the residuals in the partial differential equation, the initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms.…”

“…Thus the domain is divided into a fixed number of curvilinear quadrilaterals. These quadrilaterals are then divided into shape regular quadrilaterals of size h as has been described in [2]. Each quadrilateral is mapped onto the unit squares S = (−1, 1) × (−1, 1) and the time interval (nk, (n + 1)k) is mapped onto the interval I = (−1, 1).…”

“…In [2] a spectral element method for solving parabolic initial boundary value problems on smooth domains with Dirichlet boundary conditions using parallel computers has been proposed. The coefficients of the differential operator and data are assumed to be smooth or belong to certain Gevrey spaces.…”