Enhancing Biomechanical Simulations Based on a Posteriori Error Estimates: The Potential of Dual‐Weighted Residual‐Driven Adaptive Mesh Refinement
Sign Up to like & getrecommendations! Published in 2024 at "International Journal for Numerical Methods in Biomedical Engineering"
DOI: 10.1002/cnm.3897
Abstract: The finite‐element method (FEM) is a well‐established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with a discretisation error that can be… read more here.
Keywords: posteriori error; weighted residual; error estimates; dual weighted ... See more keywords
Approximation error estimates by noise‐injected neural networks
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DOI: 10.1002/mma.10288
Abstract: One‐hidden‐layer feedforward neural networks are described as functions having many real‐valued parameters. Approximation properties of neural networks are established (universal approximation property), and the approximation error is related to the number of parameters in the… read more here.
Keywords: approximation error; error estimates; stochastic perturbations; approximation ... See more keywords
A note on a priori $$\mathbf {L^p}$$Lp-error estimates for the obstacle problem
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DOI: 10.1007/s00211-017-0931-5
Abstract: This paper is concerned with a priori error estimates for the piecewise linear finite element approximation of the classical obstacle problem. We demonstrate by means of two one-dimensional counterexamples that the $$L^2$$L2-error between the exact… read more here.
Keywords: note priori; obstacle problem; error estimates; priori mathbf ... See more keywords
Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems
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DOI: 10.1007/s10915-018-0687-9
Abstract: In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the… read more here.
Keywords: estimates local; error; local discontinuous; solution ... See more keywords
Error Estimates for First- and Second-Order Lagrange–Galerkin Moving Mesh Schemes for the One-Dimensional Convection–Diffusion Equation
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DOI: 10.1007/s10915-024-02673-4
Abstract: A new moving mesh scheme based on the Lagrange–Galerkin method for the approximation of the one-dimensional convection–diffusion equation is studied. The mesh movement is prescribed by a discretized dynamical system for the nodal points. This… read more here.
Keywords: diffusion equation; lagrange galerkin; moving mesh; error estimates ... See more keywords
A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems
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DOI: 10.1007/s10957-018-1311-8
Abstract: We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean values of the state variable. In order to control the feasibility violation induced by the discretization,… read more here.
Keywords: semilinear parabolic; state; control; optimization ... See more keywords
Representation Results and Error Estimates for Differential Games with Applications Using Neural Networks
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DOI: 10.1007/s13235-024-00597-0
Abstract: We study deterministic optimal control problems for differential games with finite horizon. We propose new approximations of the strategies in feedback form and show error estimates and a convergence result of the value in some… read more here.
Keywords: games applications; document; optimal control; differential games ... See more keywords
Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver☆☆☆
Sign Up to like & getrecommendations! Published in 2017 at "Computer Methods in Applied Mechanics and Engineering"
DOI: 10.1016/j.cma.2017.08.019
Abstract: Abstract We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in d ≥ 1 dimensions. Our main… read more here.
Keywords: finite element; adaptive finite; error estimates; error ... See more keywords
A posteriori error estimates for Biot system using Enriched Galerkin for flow
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DOI: 10.1016/j.cma.2020.113185
Abstract: Abstract We analyze the Biot system solved with a fixed-stress split, Enriched Galerkin (EG) discretization for the flow equation, and Galerkin for the mechanics equation. Residual-based a posteriori error estimates are established with both lower… read more here.
Keywords: enriched galerkin; error; error estimates; posteriori error ... See more keywords
Explicit k-dependence for P k finite elements in W m,p error estimates: application to probabilistic laws for accuracy analysis
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DOI: 10.1080/00036811.2019.1698727
Abstract: We derive an explicit k-dependence in error estimates for Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between and finite elements, ( ), in terms of -norms. We further… read more here.
Keywords: finite elements; error estimates; explicit dependence; probabilistic laws ... See more keywords
A Posteriori Error Estimates for Lumped Mass Finite Element Method for Linear Parabolic Problems Using Elliptic Reconstruction
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DOI: 10.1080/01630563.2017.1338730
Abstract: ABSTRACT We study residual-based a posteriori error estimates for both the spatially discrete and the fully discrete lumped mass finite element approximation for parabolic problems in a bounded convex polygonal domain in ℝ2. In particular,… read more here.
Keywords: finite element; posteriori error; lumped mass; error estimates ... See more keywords