Relevant bibliographies by topics / FEM discretization

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Academic literature on the topic 'FEM discretization'

Author: Grafiati
Published: 1 March 2025

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Journal articles on the topic "FEM discretization"

1

Dryja, M., and M. Sarkis. "Additive Average Schwarz Methods for Discretization of Elliptic Problems with Highly Discontinuous Coefficients." Computational Methods in Applied Mathematics 10, no. 2 (2010): 164–76. http://dx.doi.org/10.2478/cmam-2010-0009.

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AbstractA second order elliptic problem with highly discontinuous coefficients has been considered. The problem is discretized by two methods: 1) continuous finite element method (FEM) and 2) composite discretization given by a continuous FEM inside the substructures and a discontinuous Galerkin method (DG) across the boundaries of these substructures. The main goal of this paper is to design and analyze parallel algorithms for the resulting discretizations. These algorithms are additive Schwarz methods (ASMs) with special coarse spaces spanned by functions that are almost piecewise constant with respect to the substructures for the first discretization and by piecewise constant functions for the second discretization. It has been established that the condition number of the preconditioned systems does not depend on the jumps of the coefficients across the substructure boundaries and outside of a thin layer along the substructure boundaries. The algorithms are very well suited for parallel computations.
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2

Martello, Giulia. "Discretization Analysis in FEM Models." MATEC Web of Conferences 53 (2016): 01063. http://dx.doi.org/10.1051/matecconf/20165301063.

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3

Lahtinen, Valtteri, and Antti Stenvall. "A category theoretical interpretation of discretization in Galerkin finite element method." Mathematische Zeitschrift 296, no. 3-4 (2020): 1271–85. http://dx.doi.org/10.1007/s00209-020-02456-1.

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Abstract The Galerkin finite element method (FEM) is used widely in finding approximative solutions to field problems in engineering and natural sciences. When utilizing FEM, the field problem is said to be discretized. In this paper, we interpret discretization in FEM through category theory, unifying the concept of discreteness in FEM with that of discreteness in other fields of mathematics, such as topology. This reveals structural properties encoded in this concept: we propose that discretization is a dagger mono with a discrete domain in the category of Hilbert spaces made concrete over the category of vector spaces. Moreover, we discuss parallel decomposability of discretization, and through examples, connect it to different FEM formulations and choices of basis functions.
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MARAZZINA, DANIELE, OLEG REICHMANN, and CHRISTOPH SCHWAB. "hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES." Mathematical Models and Methods in Applied Sciences 22, no. 01 (2012): 1150005. http://dx.doi.org/10.1142/s0218202512005897.

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We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.
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5

Ovchinnikov, George V., Denis Zorin, and Ivan V. Oseledets. "Robust regularization of topology optimization problems with a posteriori error estimators." Russian Journal of Numerical Analysis and Mathematical Modelling 34, no. 1 (2019): 57–69. http://dx.doi.org/10.1515/rnam-2019-0005.

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Abstract Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material). Using a finite elements discretization (FEM) of the PDE and functional we obtain an integer programming problem. Due to approximation error of the FEM discretization, optimization problem becomes mesh-depended and possess false, physically inadequate optimums, while functional value heavily depends on the fineness of discretization scheme used to compute it. To alleviate this problem, we propose regularization of given functional by error estimate of the FEM discretization. This regularization provides robustness of solutions and improves obtained functional values as well. While the idea is broadly applicable, in this paper we apply our method to the heat conduction optimization. Problems of this type are of practical importance in design of heat conduction channels, heat sinks and other types of heat guides.
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6

Schedensack, Mira. "A New Generalization of the P1 Non-Conforming FEM to Higher Polynomial Degrees." Computational Methods in Applied Mathematics 17, no. 1 (2017): 161–85. http://dx.doi.org/10.1515/cmam-2016-0031.

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AbstractThis paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.
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7

Devaud, Denis. "Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2." IMA Journal of Numerical Analysis 40, no. 4 (2019): 2717–45. http://dx.doi.org/10.1093/imanum/drz036.

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Abstract We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order $1/2$ and the Riemann–Liouville derivative of order $1/2$ with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for $hp$-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also $hp$-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of $N_t$, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial $hp$-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. $hp$-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.
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8

Yao, Lingyun, Wanyi Tian, and Fei Wu. "An Optimized Generalized Integration Rules for Error Reduction of Acoustic Finite Element Model." International Journal of Computational Methods 15, no. 07 (2018): 1850062. http://dx.doi.org/10.1142/s0219876218500627.

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In the finite element method (FEM), the accuracy in acoustic problems will deteriorate with the increasing frequency due to the “dispersion effect”. In order to minimize discretization error, a novel optimized generalized integration rules (OGIR) is introduced into FEM for the reduction of discretization error. In the present work, the adaptive genetic algorithm (AGA) is implemented to sight the optimized location of integration points. Firstly, the generalized integration rules (GIR) is used to parameterize the Gauss point location, then the relationship between the location parameterize of the integration points and discretization error is derived in detail, and the optimized location of the integration points is found through the optimization procedure, and then the OGIR–FEM is finally proposed to solve the acoustic problem. It also can be directly used to solve the optional acoustic problem, including the damped problems. Numerical example involving distorted meshes indicates that present OGIR–FEM has a superior error reducing performance in comparison with the other error reducing finite elements. These researches indicate that the proposed method can be more widely applied to solving practical acoustic problems with more accurate solutions.
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9

Zhao, Jingjun, Jingyu Xiao, and Yang Xu. "Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/857205.

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A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.
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10

Xu, Haochen. "Analyzing heat transfer in Axial Flux Permanent Magnet electrical machines: A literature review on the discretization methods-FVM and FDM." Theoretical and Natural Science 11, no. 1 (2023): 223–30. http://dx.doi.org/10.54254/2753-8818/11/20230412.

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Axial Flux Permanent Magnet (AFPM) machines have gained significant attention due to their high power density, efficiency, and compact design. However, effective heat transfer analysis is critical for optimizing their performance and reliability. This paper presents a comprehensive literature review on the application of discretization methods, specifically the Finite Volume Method (FVM) and Finite Difference Method (FDM), in the thermal analysis of AFPM machines. The fundamentals of FVM and FDM are briefly explained, followed by an exploration of their applications in AFPM machine thermal analysis. The advantages and limitations of using these methods are discussed, and a comparison between FVM and FDM is provided. Advanced discretization techniques, such as the Finite Element Method (FEM), and coupled thermal-electromagnetic analyses are also examined. The paper highlights studies that have utilized FVM or FDM in developing optimized designs and effective thermal management strategies for AFPM machines. Lastly, potential future research directions are identified, including the development of more efficient discretization methods, the incorporation of advanced materials, and the investigation of novel cooling techniques. This review offers valuable insights into the current state of research and potential future directions in the thermal analysis of AFPM machines using discretization methods.
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