172 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
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
Maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction-diffusion problem
A singularly perturbed semilinear reaction-diffusion equation, posed in the unit square, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio assumption is made. Numerical results are presented that support our theoretical estimat
A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain
The semilinear reaction-diĀ®usion equation Ā”"24u+b(x; u) = 0 with Dirichlet bound-ary conditions is considered in a convex polygonal domain. The singular perturbation parameter Īµ is arbitrarily small, and the āreduced equationā b(x, u0 (x)) = 0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation āDz + f (z) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem
- ā¦