Adaptive FEM Research Articles

Efficient P1-FEM for Any Space Dimension in Matlab

Abstract This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in R d \mathbb{R}^{d} ( d ∈ N d\in\mathbb{N} ). Although at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of the method independent of the dimension. For instance, it provides additional insights into necessary data structures and the complexity of implementations. Throughout, the focus is on an efficient realization using Matlab built-in functions and vectorization. The fast and vectorized Matlab function can be easily implemented in many other vector languages and is provided in Julia, too. The complete implementation of the adaptive FEM is given, including assembling stiffness matrix, building load vector, error estimation, and adaptive mesh-refinement. Numerical experiments underline the efficiency of our freely available code which is observed to be of a slightly more than linear complexity with respect to the number of elements when memory limits are not exceeded.

Corrigendum to: Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

Abstract Unfortunately, there is a flaw in the numerical analysis of the published version [IMA J. Numer. Anal., DOI:10.1093/imanum/drad039], which is corrected here. Neither the algorithm nor the results are affected, but constants have to be adjusted.

An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities

An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities

Adaptive mixed FEM combined with the method of characteristics for stationary convection–diffusion–reaction problems

Adaptive mixed FEM combined with the method of characteristics for stationary convection–diffusion–reaction problems

Plain convergence of goal-oriented adaptive FEM

Plain convergence of goal-oriented adaptive FEM

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

Abstract We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.

MooAFEM: An object oriented Matlab code for higher-order adaptive FEM for (nonlinear) elliptic PDEs

We present an easily accessible, object oriented code (written exclusively in Matlab) for adaptive finite element simulations in 2D. It features various refinement routines for triangular meshes as well as fully vectorized FEM ansatz spaces of arbitrary polynomial order and allows for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. Due to the object oriented programming paradigm, the code can be used easily and is readily extensible. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.

Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

We formulate and analyze a goal-oriented adaptive finite element method for a semilinear elliptic PDE and a linear goal functional. The discretization is based on finite elements of arbitrary (but fixed) polynomial degree and involves a linearized dual problem. The linearization is part of the proposed algorithm, which employs a marking strategy different to that of standard adaptive finite element methods. Moreover, unlike the well-known dual-weighted residual strategy, the analyzed error estimators are completely computable. We prove linear convergence and, for the first time in the context of goal-oriented adaptivity for nonlinear PDEs, optimal algebraic convergence rates. In particular, the analysis does not require a sufficiently fine initial mesh.

On an adaptive FEM-BEM for nonlinear transmission problem

We consider the adaptive FEM–BEM for the nonlinear transmission problem. Under the newly defined distance framework, the a posteriori and a priori error estimates are proposed and analyzed. Numerical example verifies the theoretical analysis result.

Evaluation of Optimal Design Parameters of the Geogrid Reinforced Foundation with Wraparound Ends Using Adaptive FEM

Evaluation of Optimal Design Parameters of the Geogrid Reinforced Foundation with Wraparound Ends Using Adaptive FEM

Convergence and rate optimality of adaptive multilevel stochastic Galerkin FEM

Abstract We analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in two-dimensional physical domains, with coefficients and right-hand-side functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos expansion with coefficients residing in finite element spaces associated with different locally refined meshes. Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximation spaces.

Optimal Convergence Rates for Goal-Oriented FEM with Quadratic Goal Functional

Abstract We consider a linear elliptic PDE and a quadratic goal functional. The goal-oriented adaptive FEM algorithm (GOAFEM) solves the primal as well as a dual problem, where the goal functional is always linearized around the discrete primal solution at hand. We show that the marking strategy proposed in [M. Feischl, D. Praetorius and K. G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal. 54 (2016), 3, 1423–1448] for a linear goal functional is also optimal for quadratic goal functionals, i.e., GOAFEM leads to linear convergence with optimal convergence rates.

An h-version adaptive FEM for eigenproblems in system of second order ODEs: vector Sturm-Liouville problems and free vibration of curved beams

Purpose This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions. Design/methodology/approach This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error. Findings Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance. Originality/value The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

A Novel Approach to Power Quality Analysis: Adaptive FEM Algorithm with Sparse Signal Decomposition

AbstractAn adaptive approach of discretized triangular mesh based on finite element method hybrid with type‐2 fuzzy is utilized for detection and classification of power quality events. The concept of sparse signal decomposition is utilized with extraction of disturbances from power signals. An adaptive finite element method is then applied with the concept of stiffness matrix which extracts some relevant features of power quality events that act as input to type‐2 fuzzy classifiers. Finally performance comparison is carried out, the results of which show that the proposed technique has potential to analyze disturbances with high accuracy and minimum computational overhead.

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm.

Optimal convergence behavior of adaptive FEM driven by simple [formula omitted]-type error estimators

For some Poisson-type model problem, we prove that adaptive FEM driven by the (h−h∕2)-type error estimators from Ferraz-Leite et al. (2010) leads to convergence with optimal algebraic convergence rates. Besides the implementational simplicity, another striking feature of these estimators is that they can provide guaranteed lower bounds for the energy error with known efficiency constant 1.

Convergence of Adaptive Stochastic Galerkin FEM

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.

An Adaptive FEM for the Pointwise Tracking Optimal Control Problem of the Stokes Equations

We propose and analyze a reliable and efficient a posteriori error estimator for the pointwise tracking optimal control problem of the Stokes equations. This linear-quadratic optimal control proble...

An Adaptive Least-Squares FEM for Linear Elasticity with Optimal Convergence Rates

Adaptive mesh-refining is of particular importance in computational mechanics and established here for the lowest-order locking-free least-squares finite element scheme which solely employs conforming $P_1$ approximations for the displacement and lowest-order Raviart--Thomas approximations for the stress variables. This forms a competitive discretization in particular in three-dimensional linear elasticity with traction boundary conditions although the stress approximation does not satisfy the symmetry condition exactly. The paper introduces an adaptive mesh-refining algorithm based on separate marking and exact solve with some novel explicit a posteriori error estimator and proves optimal convergence rates. The point is robustness in the sense that the crucial input parameters $\Theta$ for the Dorfler marking and $\kappa$ for the separate marking as well as the equivalence constants in the asymptotic convergence rates do not degenerate as the Lame parameter $\lambda$ tends to $\infty$.

Adaptive finite element method for parabolic equations with Dirac measure

In this paper we study the adaptive finite element method for parabolic equations with Dirac measure. Two kinds of problems with separate measure data in time and measure data in space are considered. It is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the Dirac measure, and thus fit to adaptive FEM for space discretization and variable time steps for time discretization. For both cases we use piecewise linear and continuous finite elements for the space discretization and backward Euler scheme, or equivalently piecewise constant discontinuous Galerkin method, for the time discretization, the a posteriori error estimates based on energy and L2 norms for the fully discrete problems are then derived to guide the adaptive procedure. Numerical results are provided at the end of the paper to support our theoretical findings.