Article
Mathematics, Applied
Philip Freese, Dietmar Gallistl, Daniel Peterseim, Timo Sprekeler
Summary: This paper proposes novel computational multiscale methods for solving linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The methods are based on localized orthogonal decomposition (LOD) and provide operator-adapted coarse spaces through solving localized cell problems on a fine scale, similar to numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. Rigorous error analysis of one specific approach demonstrates the applicability and accuracy of the LOD methodology for nondivergence form problems.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Timo Sprekeler, Hung Tran
Summary: This study focuses on optimal convergence rates in the periodic homogenization of linear elliptic equations, obtaining the best convergence rate in various norms and providing gradient estimates with correction terms. Numerical experiments using a specific diffusion matrix demonstrate the optimality of the obtained rates, with discussions on extending the results to nonsmooth domains and their utility in numerical homogenization.
MULTISCALE MODELING & SIMULATION
(2021)
Article
Mathematics, Applied
Omar Lakkis, Amireh Mousavi
Summary: In this article, we propose a least-squares method for recovering the gradient and possibly the Hessian of an elliptic equation in nondivergence form. By incorporating the curl-free constraint into the target functional, inf-sup stabilization is not required for the discrete spaces. We demonstrate the a priori and a posteriori convergence results using standard conforming finite elements, and validate our findings with numerical experiments involving uniform or adaptive mesh refinement.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Yalchin Efendiev, Wing Tat Leung, Wenyuan Li, Zecheng Zhang
Summary: This paper introduces a new splitting method where some degrees of freedom are handled implicitly while others are handled explicitly. Modified partial machine learning algorithms are introduced to replace the implicit solution part and proper orthogonal decomposition based model reduction is used in machine learning. Three numerical examples are presented to demonstrate the stability and accuracy of the machine learning scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics
Luis Silvestre
Summary: This article presents some properties of solutions to parabolic equations and fully nonlinear uniformly parabolic equations. By constructing examples and using numerical computations, it proves certain properties of the solutions under specific conditions. These studies are of great importance for a deeper understanding and solving problems related to parabolic equations and nonlinear uniformly parabolic equations.
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE
(2022)
Article
Mathematics, Applied
Pablo Raul Stinga, Mary Vaughan
Summary: This paper investigates the fractional powers of nondivergence form elliptic operators in bounded domains, with minimal regularity assumptions on coefficients and boundaries. The method of sliding paraboloids is developed to prove interior Harnack inequality and Holder estimates for solutions, leading to implications for solutions to the fractional problem.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2021)
Article
Computer Science, Interdisciplinary Applications
Pijus Makauskas, Mayur Pal, Vismay Kulkarni, Abhishek Singh Kashyap, Himanshu Tyagi
Summary: A neural solution methodology using feed-forward and convolutional neural networks is proposed for solving the general tensor elliptic pressure equation with discontinuous coefficients. This methodology effectively replaces traditional numerical schemes like finite-volume methods in solving single-phase flow in porous medium. The neural solution, based on machine learning algorithms, provides a faster and more accurate solution compared to finite volume schemes like TPFA or MPFA. Various test cases in 1D and 2D are presented to demonstrate the general applicability and accuracy of the proposed neural solution method.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics
Diego Maldonado
Summary: The study proves a Harnack inequality for nonnegative strong solutions to certain second-order elliptic PDEs with rough coefficients, where the degeneracies/singularities are shaped by weights in the reverse-Holder class RH infinity. Such operators include subelliptic Grushin and Grushin-like PDEs with unbounded first-order terms.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Mengya Su, Zhiyue Zhang
Summary: In this paper, we study the numerical approximation of an elliptic interface optimal control problem where the control variable acts on the interface. The discontinuity of coefficients across the interface leads to low regularity of the solution over the entire domain. To address this issue, we propose an immersed finite element method based on a uniform mesh to solve the state and adjoint equations, and discretize the control variable using a variational discretization method. Numerical experiments with complex interfaces, constrained control, no exact solution, and variable coefficients demonstrate the effectiveness of this numerical method.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
John W. Barrett, Klaus Deckelnick, Vanessa Styles
Summary: The diffuse interface approach for solving the elliptic PDE on a closed hypersurface is based on a finite element scheme and numerical quadrature, making it easy to implement compared to other methods. Error estimation is done in natural norms based on spatial grid size, interface width, and the order of the quadrature rule, with numerical test calculations confirming the error bounds.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Engineering, Multidisciplinary
Tale Bakken Ulfsby, Andre Massing, Simon Sticko
Summary: We propose a novel cut discontinuous Galerkin (CutDG) method for solving stationary advection-reaction problems on surfaces embedded in Rd. The approach involves embedding the surface into a full-dimensional background mesh and using discontinuous piecewise polynomials as test and trial functions. By introducing a suitable stabilization technique, we are able to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. Numerical examples validate our theoretical findings.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Interdisciplinary Applications
Dietmar Gallistl, Timo Sprekeler, Endre Suli
Summary: The paper introduces a mixed finite element method for approximating the periodic strong solution to the fully nonlinear second-order Hamilton-Jacobi-Bellman equation with coefficients satisfying the Cordes condition. The second part of the paper focuses on numerically homogenizing such equations and approximating the effective Hamiltonian. Numerical experiments demonstrate the effectiveness of the approximation scheme for the effective Hamiltonian and the numerical solution for the homogenized problem.
MULTISCALE MODELING & SIMULATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Wenjia Jing, Yiping Zhang
Summary: This paper studies the quantitative homogenization of linear second order elliptic equations in nondivergence form, which have highly oscillating periodic diffusion coefficients and large drifts. By transforming the equation into divergence form through a centering condition and invariant measures, the modified diffusion coefficients are obtained without drift. Applying existing quantitative homogenization results, quantitative estimates for equations in nondivergence form with large drifts, such as convergence rates and uniform Lipschitz regularity, can be obtained.
MULTISCALE MODELING & SIMULATION
(2023)
Article
Mathematics, Applied
Andrew T. Barker, Andrei Draganescu
Summary: The study introduces an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear-quadratic optimization problem constrained by an elliptic partial differential equation. The construction of the preconditioner relies on a standard AMG infrastructure built for solving the forward elliptic equation, allowing for broad applicability to problems with unstructured grids, complex geometry, and varying coefficients. The method is implemented using the Hypre package and numerical examples are provided.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Carsten Carstensen, Rui Ma
Summary: This paper establishes the convergence of adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite elliptic problems in three dimensions with piecewise Lipschitz continuous coefficients. The adaptive algorithm with collective Dorfler marking is utilized to measure the error in L-2 norms. The axioms of adaptivity apply to this setting and guarantee the rate optimality for Raviart-Thomas and Brezzi-Douglas-Marini finite elements of any order for sufficiently small initial mesh-sizes and bulk parameter.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Computer Science, Theory & Methods
Wolfgang Dahmen, Ronald DeVore, Lars Grasedyck, Endre Suli
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2016)
Editorial Material
Physics, Applied
Alpha A. Lee, Andreas Muench, Endre Sueli
APPLIED PHYSICS LETTERS
(2016)
Article
Mathematics, Applied
John W. Barrett, Endre Suli
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2018)
Article
Mechanics
Josef Malek, Vit Prusa, Tomas Skrivan, Endre Suli
Article
Mathematics, Applied
Graham Baird, Endre Suli
Summary: This paper presents a numerical scheme for a mixed discrete-continuous fragmentation equation, ensuring mass conservation and nonnegativity preservation. By utilizing operator semigroups, it is shown that the weak solutions are unique and classical, with a bound derived for the error induced by truncation.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Mathematics, Applied
E. L. L. Y. A. L. KAWECKI, T. I. M. O. SPREKELER
Summary: In the first part, the paper studies the discontinuous Galerkin (DG) and C(0) interior penalty (C(0)-IP) finite element approximation of the periodic strong solution to the fully nonlinear second-order Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation with coefficients satisfying the Cordes condition. Well-posedness is proved and abstract a posteriori and a priori analyses are performed, which apply to a wide family of numerical schemes. The second part focuses on the numerical approximation to the effective Hamiltonian of ergodic HJBI operators using DG/C(0)-IP finite element approximations. Numerical experiments are provided to demonstrate the performance of the numerical schemes.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
I. P. A. Papadopoulos, E. Sull
Summary: This article extends the topology optimization model proposed by Borrvall and Petersson (2003) and proves novel regularity results. The research shows that, for a given infinite-dimensional problem, there exists a sequence of finite element solutions that strongly converge to the local minimizer and satisfy the necessary first-order optimality conditions. Additionally, the article provides the first numerical investigation into convergence rates.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Miroslav Bulicek, Victoria Patel, Endre Suli, Yasemin Sengul
Summary: This article discusses a system of evolutionary equations that describe certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The existence and uniqueness of a global-in-time large-data weak solution is established for a large class of implicit constitutive relations. The focus is then placed on the class of limiting strain models, where a technical difficulty arises due to the integrability of the Cauchy stress. However, a satisfactory existence theory can still be provided as long as the initial data have finite elastic energy and the boundary data fulfill natural compatibility conditions.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Philip Freese, Dietmar Gallistl, Daniel Peterseim, Timo Sprekeler
Summary: This paper proposes novel computational multiscale methods for solving linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The methods are based on localized orthogonal decomposition (LOD) and provide operator-adapted coarse spaces through solving localized cell problems on a fine scale, similar to numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. Rigorous error analysis of one specific approach demonstrates the applicability and accuracy of the LOD methodology for nondivergence form problems.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Timo Sprekeler, Hung Tran
Summary: This study focuses on optimal convergence rates in the periodic homogenization of linear elliptic equations, obtaining the best convergence rate in various norms and providing gradient estimates with correction terms. Numerical experiments using a specific diffusion matrix demonstrate the optimality of the obtained rates, with discussions on extending the results to nonsmooth domains and their utility in numerical homogenization.
MULTISCALE MODELING & SIMULATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Dietmar Gallistl, Timo Sprekeler, Endre Suli
Summary: The paper introduces a mixed finite element method for approximating the periodic strong solution to the fully nonlinear second-order Hamilton-Jacobi-Bellman equation with coefficients satisfying the Cordes condition. The second part of the paper focuses on numerically homogenizing such equations and approximating the effective Hamiltonian. Numerical experiments demonstrate the effectiveness of the approximation scheme for the effective Hamiltonian and the numerical solution for the homogenized problem.
MULTISCALE MODELING & SIMULATION
(2021)
Article
Physics, Fluids & Plasmas
Mark Dostalik, Josef Malek, Vit Prusa, Endre Suli
