5 research outputs found
Adaptive energy minimisation for hp-finite element methods
This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner. Specifically, we outline a new approach in the context of hp-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new hp-refinement technique for both one- and two-dimensional problems
Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii Equation
We present an effective adaptive procedure for the numerical approximation of
the steady-state Gross-Pitaevskii equation. Our approach is solely based on
energy minimization, and consists of a combination of gradient flow iterations
and adaptive finite element mesh refinements. Numerical tests show that this
strategy is able to provide highly accurate results, with optimal convergence
rates with respect to the number of freedom
An -adaptive strategy based on locally predicted error reductions
We introduce a new -adaptive strategy for self-adjoint elliptic boundary
value problems that does not rely on using classical a posteriori error
estimators. Instead, our approach is based on a generally applicable prediction
strategy for the reduction of the energy error that can be expressed in terms
of local modifications of the degrees of freedom in the underlying discrete
approximation space. The computations related to the proposed prediction
strategy involve low-dimensional linear problems that are computationally
inexpensive and highly parallelizable. The mathematical building blocks for
this new concept are first developed on an abstract Hilbert space level, before
they are employed within the specific context of -type finite element
discretizations. For this particular framework, we discuss an explicit
construction of -enrichments and -refinements by means of an appropriate
constraint coefficient technique that can be employed in any dimensions. The
applicability and effectiveness of the resulting -adaptive strategy is
illustrated with some - and -dimensional numerical examples
An hp-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems
In this paper we develop an hp-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an hp-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust hp-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples
Recommended from our members
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs