Kendall Atkinson

Kendall Atkinson, David Chien, Olaf Hansen

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.

Kendall Atkinson, Olaf Hansen

Consider being given a mapping ϕfrom the unit sphere S^{d-1}, d>2, to the smooth boundary of a simply-connected region Ωin R^d. We consider the problem of constructing an extension Φfrom the unit ball B_d to Ω. The mapping is required to be 1-1 and continuously differentiable with a nonsingular Jacobian matrix. We discuss ways of obtaining initial guesses for such a mapping Φand of then improving it by an iteration method.

Kendall Atkinson, Olaf Hansen, David Chien

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region Ω for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both R^2 and R^3.

Kendall Atkinson, David Chien, Olaf Hansen

Let $Ω$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partialΩ$ is smooth. Consider solving an elliptic partial differential equation $Lu=f$ over $Ω$ with zero Dirichlet boundary value. The function $f$ is a nonlinear function of the solution $u$. The problem is converted to an equivalent\ elliptic problem over the open unit ball $\mathbb{B}^{d}$ in $\mathbb{R}^{d}$, say $\widetilde{L}\widetilde{u}=\widetilde{f}$. Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $\widetilde{u}_{n}$ of degree $\leq n$ that is convergent to $\widetilde{u}$. The transformation from $Ω$ to $\mathbb{B}^{d}$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}\left( \overline{Ω}\right) $ and assuming $\partialΩ$ is a $C^{\infty}$ boundary, the convergence of $\left\Vert \widetilde{u}-\widetilde{u}_{n}\right\Vert _{H^{1}% }$ \ to zero is faster than any power of $1/n$. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to $-Δu+γu=f$ with a zero Neumann boundary condition is also presented.

Kendall Atkinson, Olaf Hansen

Let $Ω$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partialΩ$ is smooth. Consider solving the eigenvalue problem $Lu=λu$ for an elliptic partial differential operator $L$ over $Ω$ with zero values for either Dirichlet or Neumann boundary conditions. We propose, analyze, and illustrate a 'spectral method' for solving numerically such an eigenvalue problem. This is an extension of the methods presented earlier in [5],[6].

Kendall Atkinson, David Chien, Olaf Hansen

Let $Ω$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partialΩ$ is smooth. Consider solving an 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 Galerkin method is used to create a convergent sequence of multivariate polynomials $u_{n}$ of degree $\leq n$ 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 $u\in C^{\infty}(\overlineΩ) $ and assuming $\partialΩ$ is a $C^{\infty}$ boundary, the convergence of $\Vert u-u_{n}\Vert_{H^{1}}$ to zero is faster than any power of $1/n$. Numerical examples in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ show experimentally an exponential rate of convergence.