3 research outputs found
Accuracy of spectral element method for wave, parabolic and Schr\"{o}dinger equations
The spectral element method constructed by the () continuous
finite element method with -point Gauss-Lobatto quadrature on
rectangular meshes is a popular high order scheme for solving wave equations in
various applications. It can also be regarded as a finite difference scheme on
all Gauss-Lobatto points. We prove that this finite difference scheme is
-order accurate in discrete 2-norm for smooth solutions. The same proof
can be extended to the spectral element method solving linear parabolic and
Schr\"odinger equations. The main result also applies to the spectral element
method on curvilinear meshes that can be smoothly mapped to rectangular meshes
on the unit square
Superconvergence of - finite element method for elliptic equations with approximated coefficients
We prove that the superconvergence of - finite element method at
the Gauss Lobatto quadrature points still holds if variable coefficients in an
elliptic problem are replaced by their piecewise Lagrange interpolant at
the Gauss Lobatto points in each rectangular cell. In particular, a fourth
order finite difference type scheme can be constructed using - finite
element method with approximated coefficients
Superconvergence of high order finite difference schemes based on variational formulation for elliptic equations
The classical continuous finite element method with Lagrangian basis
reduces to a finite difference scheme when all the integrals are replaced by
the Gauss-Lobatto quadrature. We prove that this finite
difference scheme is -th order accurate in the discrete 2-norm for an
elliptic equation with Dirichlet boundary conditions, which is a
superconvergence result of function values