876 research outputs found
Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions
Assuming is a ball in , we analyze the positive
solutions of the problem that branch out from the constant solution as grows from to
. The non-zero constant positive solution is the unique positive
solution for close to . We show that there exist arbitrarily many
positive solutions as (in particular, for supercritical exponents)
or as for any fixed value of , answering partially a
conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for
and so that a given number of solutions exist. The geometrical properties
of those solutions are studied and illustrated numerically. Our simulations
motivate additional conjectures. The structure of the least energy solutions
(among all or only among radial solutions) and other related problems are also
discussed.Comment: 37 pages, 24 figure
Adaptive Pseudo-Transient-Continuation-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we investigate the application of pseudo-transient-continuation
(PTC) schemes for the numerical solution of semilinear elliptic partial
differential equations, with possible singular perturbations. We will outline a
residual reduction analysis within the framework of general Hilbert spaces,
and, subsequently, employ the PTC-methodology in the context of finite element
discretizations of semilinear boundary value problems. Our approach combines
both a prediction-type PTC-method (for infinite dimensional problems) and an
adaptive finite element discretization (based on a robust a posteriori residual
analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
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
An -Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems
In this paper we develop an -adaptive procedure for the numerical
solution of general, semilinear elliptic boundary value problems in 1d, with
possible singular perturbations. Our approach combines both a prediction-type
adaptive Newton method and an -version adaptive finite element
discretization (based on a robust a posteriori residual analysis), thereby
leading to a fully -adaptive Newton-Galerkin scheme. Numerical experiments
underline the robustness and reliability of the proposed approach for various
examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
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