Convergence of an adaptive hp finite element strategy in higher space-dimensions

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems

We develop a new analysis for residual-type aposteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber as our model problem. We employ a classical conforming Galerkin discretization by using hp-finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), pp.1871-1914], Melenk and Sauter introduced an hp-finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only degrees of freedom, where denotes the spatial dimension. In the present paper, we will introduce an aposteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the aposteriori estimates would be amplified by a factor

Numerical Optimization of a Waveguide Transition Using Finite Element Beam Propagation

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Implementation of the coupled two-mode phase field crystal model with Cahn–Hilliard for phase-separation in battery electrode particles

In this article, we present the behavior of two‐mode phase field crystal (2MPFC) method under a concentration dependent deformation. A mixed finite element formulation is proposed for the 2MPFC method that solves a 10th‐order parabolic equation. Lithium concentration diffusion in the electrode particle is captured by the Cahn–Hilliard (CH) equation and the host electrode material, Li$_{x}$Mn$_{2}$O$_{4}$ (LMO), which has a face‐centered cubic (fcc) lattice structure, is modeled using 2MPFC. The coupling between 2MPFC and CH models brings about the concentration dependent deformation in the polycrystalline LMO electrode particle. The atomistic dynamics is assumed to operate on a faster time‐scale compared to the diffusion of lithium, thereby both the 2MPFC and CH models evolve on two different time‐scales. The coupled 2MPFC–CH system models the diffusion induced grain boundary migration in LMO capturing the charging and discharging state of the battery

Study on an hp‐Adaptive Finite Element Solver for a Chemo‐Mechanical Battery Particle Model

We consider a Cahn–Hilliard-type phase-field model for phase separation and large deformations in battery electrode particles. For the numerical solution we employ an hp-adaptive finite element solution algorithm coupled to a variable-step, variable-order time stepping scheme. Numerical experiments show the adaptive meshing and distribution of the locally varying polynomial degrees of the finite element method. In particular, for a sufficient large range of polynomial degrees we achieve significant computational savings compared to an h-adaptive algorithm

Space-Time Discontinuous Galerkin Methods for Weak Solutions of Hyperbolic Linear Symmetric Friedrichs Systems

We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations in space and time with full upwind, and we show primal and dual consistency. Stability and convergence estimates are provided with respect to a mesh-dependent DG norm which includes the L$_2$ norm at final time. Numerical experiments confirm that the a priori results are of optimal order also for solutions with low regularity, and we show that the error in the DG norm can be closely approximated with a residual-type error indicator

Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems. Revised March 2016

We introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov-Galerkin scheme in time. We show well-posedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goal-oriented dual-weighted error estimation. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for the linear transport equation and the Maxwell equation in 2D underline the effciency of the overall adaptive solution process

Space-time discontinuous Galerkin methods for weak solutions of hyperbolic linear symmetric Friedrichs systems

We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations in space and time with full upwind, and we show primal and dual consistency. Stability and convergence estimates are provided with respect to a mesh-dependent DG norm which includes the $L_2$ norm at final time. Numerical experiments confirm that the a piori results are of optimal order also for solutions with low regularity, and we show that the error in the DG norm can be closely approximated with a residual-type error indicator

Parallel space-time solutions for the linear visco-acoustic and visco-elastic wave equation

We present parallel adaptive results for a discontinuous Galerkin space-time discretization for acoustic and elastic waves with attenuation. The method is based on $p$-adaptive polynomial discontinuous ansatz and test spaces and a first-order formulation with full upwind fluxes. Adaptivity is controlled by dual-primal error estimation, and the full linear system is solved by a Krylov method with space-time multilevel preconditioning. The discretization and solution method is introduced in Dörfler-Findeisen-Wieners (Comput. Meth. Appl. Math. 2016) for general linear hyperbolic systems and applied to acoustic and elastic waves in Dörfler-Findeisen-Wieners-Ziegler (Radon Series Comp. Appl. Math. 2019); attenuation effects were included in Ziegler (PhD thesis 2019, Karlsruhe Institute of Technology). Here, we consider the evaluation of this method for a benchmark configuration in geophysics, where the convergence is tested with respect to seismograms. We consider the scaling on parallel machines and we show that the adaptive method based on goal-oriented error estimation is able to reduce the computational effort substantially