Olaf Hansen

A spectral method for elliptic equations: the Dirichlet problem

Let Ω be an open, simply connected, and bounded region in ℝd, d ≥ 2, and assume its boundary

A spectral method for elliptic equations: the Neumann problem

\partial\Omega

Evaluating polynomials over the unit disk and the unit ball

is smooth. Consider solving an elliptic partial differential equation Lu = f over Ω with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B; and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials un of degree ≤ n that is convergent to u. The transformation from Ω to B requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For

A spectral method for parabolic differential equations

u\in C^{\infty}( \overline{\Omega})

ANALYSIS OF A SPHERICAL HARMONICS EXPANSION MODEL OF PLASMA PHYSICS

and assuming

A spectral method for nonlinear elliptic equations

\partial\Omega

A spectral method for an elliptic equation with a nonlinear Neumann boundary condition

is a C ∞  boundary, the convergence of

A Spectral Method for the Biharmonic Equation

\left\Vert u-u_{n}\right\Vert _{H^{1}}

On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

to zero is faster than any power of 1/n. Numerical examples in ℝ2 and ℝ3 show experimentally an exponential rate of convergence.

SOLVING THE NONLINEAR POISSON EQUATION ON THE UNIT DISK

Let Ω be an open, simply connected, and bounded region in ℝd, d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving the elliptic partial differential equation − Δu + γu = f over Ω with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable, and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.