An hphp-Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems

In this paper we develop an $hp$-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 $hp$-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully $hp$-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

Robust equilibration a posteriori error estimation for convection-diffusion-reaction problems

We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H (div, Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved. Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases

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

An a-posteriori adaptive mesh technique for singularly perturbed convection-diffusion problems with a moving interior layer

We study numerical approximations for a class of singularly perturbed problems of convection-diffusion type with a moving interior layer. In a domain (a segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not~converge $eps$-uniformly in the uniform norm, even under the condition $N^{-1}+N_0^{-1}approx eps$, where $eps$ is the perturbation parameter and $N$ and $N_0$ denote the number of mesh points with respect to $x$ and $t$. In the case of rectangular meshes which are ({it a~priori,} or {it a~posteriori,}) locally refined in the transition layer, there are no schemes that convergence uniformly in $eps$ even under the {it very restrictive,} condition $N^{-2}+N_0^{-2} approx eps$. However, the condition for convergence can be {it essentially weakened} if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an {it a~priori,}, or an {it a~posteriori,} adaptive mesh technique. Here we construct a scheme on {it a~posteriori,} adaptive meshes (based on the gradient of the solution), whose solution converges `almost $eps$-uniformly', viz., under the condition $N^{-1}=o(eps^{ u})$, where $u>0$ is an arbitrary number from the half-open interval $(0,1]$