Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Spectral Method for Nonlinear Elliptic Equations   Let $\Omega$ be an open, simply connected, and bounded region in
$\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth.
Consider solving an elliptic partial differential equation $Lu=f$ over $\Omega$
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 $\Omega$ 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{\Omega }\right) $ and assuming $\partial\Omega$ 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 $-\Delta u+\gamma u=f$ with a zero
Neumann boundary condition is also presented.

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The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: math-ph
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org