More Like this

1. A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

   https://doi.org/10.1090/cams/39  

   Brenner, Susanne ; Sung, Li-yeng ; Tan, Zhiyu ; Zhang, Hongchao ( October 2024 , Communications of the American Mathematical Society)

   We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in${\mathbb {R}}^2$. It is based on an isoparametric$C^0$finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions.A priorianda posteriorierror estimates together with corroborating numerical results are presented.

    

   more » « less
2. A C 0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

   https://doi.org/10.1002/num.22359  

   Bhandari, Chandi ; Hoppe, Ronald H. W. ; Kumar, Rahul ( February 2019 , Numerical Methods for Partial Differential Equations)

   We consider the numerical solution of a fourth‐order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth‐order parabolic equation, we perform an implicit discretization in time and a C⁰Interior Penalty Discontinuous Galerkin (C⁰IPDG) discretization in space. The C⁰IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C⁰IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C⁰IPDG method will be shown in the second part of this paper.

    

   more » « less
3. A C 0 interior penalty discontinuous Galerkin Method for fourth‐order total variation flow. II: Existence and uniqueness

   https://doi.org/10.1002/num.22365  

   Bhandari, Chandi ; Hoppe, Ronald H. W. ; Kumar, Rahul ( February 2019 , Numerical Methods for Partial Differential Equations)

   We prove the existence and uniqueness of a solution of a C⁰Interior Penalty Discontinuous Galerkin (C⁰IPDG) method for the numerical solution of a fourth‐order total variation flow problem that has been developed in part I of the paper. The proof relies on a nonlinear version of the Lax‐Milgram Lemma. It requires to establish that the nonlinear operator associated with the C⁰IPDG approximation is Lipschitz continuous and strongly monotone on bounded sets of the underlying finite element space.

    

   more » « less
4. convergence of a homotopy finite element method for computing steady states of Burgers' equation

   https://doi.org/10.1051/m2an/2018046  

   Yang, Yong ; Hao, Wenrui ( January 2018 , ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, the convergence of a homotopy method (1.1) for solving the steady state problem of Burgers’ equation is considered. When ν is fixed, we prove that the solution of (1.1) converges to the unique steady state solution as epsilon → 0, which is independent of the initial conditions. Numerical examples are presented to confirm this conclusion by using the continuous finite element method. In contrast, when ν = epsilon → 0, numerically we show that steady state solutions obtained by (1.1) indeed depend on initial conditions. 

   more » « less
5. A convexity enforcing C0 interior penalty method for the Monge–Ampère equation on convex polygonal domains

   Brenner, Susanne C ; Sung, Li-yeng ; Tan, Zhiyu ; Zhang, Hongchao ( June 2021 , Numerische Mathematik)

   We design and analyze a C0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions. 

   more » « less