3,287 research outputs found
Fully Adaptive Newton-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we develop an adaptive procedure for the numerical solution of
general, semilinear elliptic problems with possible singular perturbations. Our
approach combines both a prediction-type adaptive Newton method and an adaptive
finite element discretization (based on a robust a posteriori error analysis),
thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples
The Method of Fundamental Solutions for Direct Cavity Problems in EIT
The Method of Fundamental Solutions (MFS) is an effective technique for solving linear elliptic partial differential equations, such as the Laplace and Helmholtz equation. It is a form of indirect boundary integral equation method and a technique that uses boundary collocation or boundary fitting. In this paper the MFS is implemented to solve A numerically an inverse problem which consists of finding an unknown cavity within a region of interest based on given boundary Cauchy data. A range of examples are used to demonstrate that the technique is very effective at locating cavities in two-dimensional geometries for exact input data. The technique is then developed to include a regularisation parameter that enables cavities to be located accurately and stably even for noisy input data
Statistical exponential formulas for homogeneous diffusion
Let denote the -homogeneous -Laplacian, for . This paper proves that the unique bounded, continuous viscosity
solution of the Cauchy problem \left\{ \begin{array}{c} u_{t} \ - \ (
\frac{p}{ \, N + p - 2 \, } ) \, \Delta^{1}_{p} u ~ = ~ 0 \quad \mbox{for}
\quad x \in \mathbb{R}^{N}, \quad t > 0 \\ \\ u(\cdot,0) ~ = ~ u_{0} \in BUC(
\mathbb{R}^{N} ) \end{array} \right. is given by the exponential formula
where the statistical operator is defined by with , when and by with , when . Possible extensions to problems with Dirichlet boundary conditions and to
homogeneous diffusion on metric measure spaces are mentioned briefly
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