Article
Engineering, Multidisciplinary
Troy Shilt, Patrick J. O'Hara, Jack J. McNamara
Summary: The article explores the alleviation of spurious oscillations introduced by traditional finite element methods in advection dominated problems through a generalized finite element formulation. This method is demonstrated to effectively capture boundary layer development and provide smooth numerical solutions with improved error levels compared to traditional formulations. Insights into the improvements offered by the generalized finite element method are further illuminated through a consistent decomposition of the variational multiscale method for comparison with classical stabilized methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Engineering, Biomedical
Yatao Liu
Summary: This study develops a mathematical model to investigate the formation and growth of atherosclerosis plaque. The results suggest that inflammation and material accumulation are more likely to occur at low shear stresses, and the plaque growth is influenced by the wall porosity.
BIOMEDICAL SIGNAL PROCESSING AND CONTROL
(2023)
Article
Mathematics, Applied
Huan Liu, Xiangcheng Zheng, Hong Wang, Hongfei Fu
Summary: In this paper, a finite element method is developed and analyzed for solving a one-dimensional two-sided time-dependent space-fractional diffusion equation. The regularity of the solution and the property of an elliptic projection are proven to derive the error estimate for the time-dependent evolution problems. Compared with existing finite element methods, the developed numerical analysis techniques have relaxed assumptions on the variable coefficient and potential extensions to higher-dimensional problems.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Elena Bachini, Matthew W. Farthing, Mario Putti
Summary: In this study, a finite element method for PDEs on surfaces was developed based on a geometrically intrinsic formulation. The method was evaluated for steady and transient problems involving diffusion and advection-dominated regimes, showing expected convergence rates and good performance compared to established finite element methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
B. V. Rathish Kumar, Manisha Chowdhury
Summary: In this paper, a fully coupled system of transient Navier-Stokes fluid flow model and unsteady variable coefficient advection-diffusion-reaction transport model is studied using subgrid multiscale stabilized finite element method. The stabilized variational formulation of the coupled system and standard expressions for the stabilization parameters are proposed by considering the algebraic approach of approximating the subscales. The time dependence of the unknown subgrid scales is considered. The stability analysis and error estimates for the stabilized finite element scheme are conducted to validate the performance and credibility of the proposed method through various numerical experiments.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Xiangcheng Zheng, Huan Liu, Hong Wang, Hongfei Fu
Summary: Finite element approximations to variable-coefficient two-sided space-fractional advection-reaction-diffusion equations in three space dimensions are analyzed. Weak coercivity of the bilinear form is proven via Garding's inequality, leading to the optimal-order error estimate of the finite element method. The method is extended to analyze time-dependent problems, with numerical experiments validating the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Jinhong Jia, Hong Wang, Xiangcheng Zheng
Summary: A fast numerical method is developed for a variably distributed-order time-fractional diffusion equation modeling anomalous diffusion with uncertainties in inhomogeneous medium. The method has the same accuracy as traditional schemes but requires less computations and storage. Additionally, a fast divide and conquer algorithm is designed to reduce computational complexity when solving the linear system.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Daxin Nie, Jing Sun, Weihua Deng
Summary: This paper studies the numerical method for solving the stochastic fractional diffusion equation driven by fractional Gaussian noise. The regularity estimate of the mild solution and the fully discrete scheme with finite element approximation in space and backward Euler convolution quadrature in time are presented using the operator theoretical approach. The convergence rates in time and space are obtained, showing the relationship between the regularity of noise and convergence rates.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Engineering, Multidisciplinary
B. V. Rathish Kumar, Manisha Chowdhury
Summary: This study presents a new subgrid multiscale stabilized formulation for non-Newtonian Casson fluid flow model tightly coupled with variable diffusion coefficients Advection-Diffusion-Reaction equation (V ADR). The stabilized formulation simplifies the equations by eliminating unresolvable scales and involving only the coarse scale solution. It investigates the relationship between the Casson viscosity coefficient and solute mass concentration, and the stability and convergence properties of the finite element solution.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Hu Chen, Yue Wang, Hongfei Fu
Summary: The sharp H-1 norm error estimate of a finite difference method for the two-dimensional time-fractional diffusion equation is established, with the Caputo time-fractional derivative term approximated by the Alikhanov scheme on a graded mesh. The temporal convergence order of the fully scheme is O(N- min{r alpha,N-2}), and the error bound remains stable as a approaches 1(-). The theoretical analysis utilizes an improved discrete fractional Gronwall inequality, and the correctness of the theoretical result is verified through numerical experiments.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Shubin Fu, Eric T. Chung, Guanglian Li
Summary: The proposed method is an Edge Multiscale Finite Element Method (EMsFEM) based on an Interior Penalty Discontinuous Galerkin (IPDG) formulation for solving heterogeneous Helmholtz problems with large wavenumber. A novel local multiscale space is constructed to capture the local behavior of wave propagation and media information. The key aspect of the method is selecting appropriate Dirichlet data and utilizing an IPDG formulation to generate a sparse linear system, reducing computational complexity.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Zhongguo Zhou, Tongtong Hang, Hao Pan, Yan Wang
Summary: This paper investigates the problem of solving three-dimensional two-sided space-fractional advection-diffusion equation using the mass conservative characteristic finite difference method. By combining the operator splitting technique with the upwind piecewise parabola method, stable and first-order convergent numerical solutions are obtained. Numerical experiments are conducted to validate the theoretical analysis.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Engineering, Civil
Youssef I. Hafez
Summary: This study developed a new numerical model to investigate the transport of radon under different conditions and verified with field data, explaining the interaction between different mechanisms of radon, proposing transport numbers and a transport similarity concept.
JOURNAL OF HYDROLOGY
(2022)
Article
Mathematics, Applied
Cunyun Nie, Shi Shu, Menghuan Liu
Summary: This paper presents a novel monotone finite volume element scheme for the diffusion problem on triangular grids. The scheme introduces fictitious triangular elements and extends the diffusive tensor and solution continuously to these elements. Additionally, a new nonlinear two-point flux formulation is obtained for some control volume edges. The paper proves the monotonicity of the scheme under certain conditions and verifies its accuracy and stability through numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Cunyun Nie, Haiyuan Yu
Summary: In this paper, we propose a monotone finite volume element scheme for diffusion problems on triangular grids. A new nonlinear two-point flux formulation is used to approximate the diffusion term, and the scheme is proven to be monotone under certain conditions. Numerical experiments validate the accuracy, stability, and monotonicity of the proposed scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Kathrin Smetana, Mario Ohlberger
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2017)
Article
Mathematics, Applied
Andreas Buhr, Christian Engwer, Mario Ohlberger, Stephan Rave
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2017)
Article
Mathematics, Interdisciplinary Applications
Mario Ohlberger, Barbara Verfuerth
MULTISCALE MODELING & SIMULATION
(2018)
Article
Computer Science, Interdisciplinary Applications
Julian Feinauer, Simon Hein, Stephan Rave, Sebastian Schmidt, Daniel Westhoff, Jochen Zausch, Oleg Iliev, Arnulf Latz, Mario Ohlberger, Volker Schmidt
JOURNAL OF COMPUTATIONAL SCIENCE
(2019)
Article
Mathematics, Applied
Stefan Hain, Mario Ohlberger, Mladjan Radic, Karsten Urban
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2019)
Article
Mathematics, Applied
Peter Bastian, Markus Blatt, Andreas Dedner, Nils-Arne Dreier, Christian Engwer, Rene Fritze, Carsten Graeser, Christoph Grueninger, Dominic Kempf, Robert Kloefkorn, Mario Ohlberger, Oliver Sander
Summary: This paper introduces the basic concepts and module structure of DUNE, as well as the recent developments and changes since its first release in 2007. It also discusses advanced features of DUNE, such as domain coupling, grid modifications, high order discretizations, and node level performance, and briefly touches on the future development directions of the framework.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Tim Keil, Luca Mechelli, Mario Ohlberger, Felix Schindler, Stefan Volkwein
Summary: In this paper, new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameters constraints are proposed and rigorously analyzed. The approach utilizes successively enriched Reduced Basis surrogate models and the projected BFGS method to solve each Trust-Region sub-problem. A non-conforming dual (NCD) approach is also introduced to improve the standard RB approximation. The resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm demonstrates a significant reduction in computational demand for large scale or multi-scale PDE constrained optimization problems.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Mathematics, Applied
Tobias Leibner, Mario Ohlberger
Summary: The article presents a new numerical method for kinetic equations based on a variable transformation of the moment approximation, overcoming limitations of traditional approaches and offering higher computational efficiency and feasibility.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Mathematics, Applied
Tim Keil, Hendrik Kleikamp, Rolf J. Lorentzen, Micheal B. Oguntola, Mario Ohlberger
Summary: In this article, we introduce an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, with a focus on polymer flooding. The framework utilizes an adaptive training procedure to directly approximate the input-output relationship of the underlying optimization problem, and emphasizes on constructing an accurate surrogate model related to the optimization path. The certified results are obtained through true evaluations of the objective function. Numerical experiments are conducted to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
M. Landstorfer, M. Ohlberger, S. Rave, M. Tacke
Summary: In this contribution, a modelling and simulation framework for parametrised lithium-ion battery cells is presented. A continuum model for a general intercalation battery cell is derived based on non-equilibrium thermodynamics. The reduced basis method is employed to efficiently evaluate the resulting parameterised non-linear system of partial differential equations. The modelling framework is particularly suitable for investigating and quantifying degradation effects of battery cells.
EUROPEAN JOURNAL OF APPLIED MATHEMATICS
(2023)
Proceedings Paper
Computer Science, Interdisciplinary Applications
Pavel Gavrilenko, Bernard Haasdonk, Oleg Iliev, Mario Ohlberger, Felix Schindler, Pavel Toktaliev, Tizian Wenzel, Maha Youssef
Summary: We propose an integrated approach that combines simulated data from full order discretization and projection-based Reduced Basis reduced order models for training machine learning approaches, particularly Kernel Methods, to achieve fast and reliable predictive models for the chemical conversion rate in reactive flows with varying transport regimes.
LARGE-SCALE SCIENTIFIC COMPUTING (LSSC 2021)
(2022)
Proceedings Paper
Computer Science, Interdisciplinary Applications
Tim Keil, Mario Ohlberger
Summary: Projection based model order reduction is a mature technique for simulating parameterized systems. However, challenges remain for problems where the solution manifold cannot be approximated well by linear subspaces.
LARGE-SCALE SCIENTIFIC COMPUTING (LSSC 2021)
(2022)
Article
Mathematics, Interdisciplinary Applications
Mario Ohlberger, Ben Schweizer, Maik Urban, Barbara Verfuerth
NETWORKS AND HETEROGENEOUS MEDIA
(2020)
Proceedings Paper
Thermodynamics
Mario Ohlberger, Felix Schindler
FINITE VOLUMES FOR COMPLEX APPLICATIONS VIII-HYPERBOLIC, ELLIPTIC AND PARABOLIC PROBLEMS
(2017)
Article
Mathematics, Applied
Mario Ohlberger, Barbara Verfuerth
