A C 0 conforming dg finite element method for biharmonic equations on triangle/tetrahedron

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiu Ye ◽  
Shangyou Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Optimal Order ◽  
Biharmonic Equations ◽  
Order Error ◽  
Simple Formulation ◽  
Piecewise Polynomials ◽  
Second Gradient ◽  
Strong Gradient ◽  
Element Method

Abstract A C 0 conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C 0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C 1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence.

scholarly journals Expanded Mixed Finite Element Method for the Two-Dimensional Sobolev Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Qing-li Zhao ◽  
Zong-cheng Li ◽  
You-zheng Ding
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Two Dimensional ◽  
Sobolev Equation ◽  
Order Error ◽  
Expanded Mixed Finite Element ◽  
Element Method

Expanded mixed finite element method is introduced to approximate the two-dimensional Sobolev equation. This formulation expands the standard mixed formulation in the sense that three unknown variables are explicitly treated. Existence and uniqueness of the numerical solution are demonstrated. Optimal order error estimates for both the scalar and two vector functions are established.

scholarly journals Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

2021 ◽  
Author(s):  
şuayip toprakseven ◽  
Peng Zhu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Reaction ◽  
Stability And Convergence ◽  
Optimal Order ◽  
Galerkin Finite Element Method ◽  
Weak Galerkin ◽  
Order Error ◽  
Galerkin Finite Element ◽  
Element Method

In this paper, a modified weak Galerkin finite element method on Shishkin mesh has been developed and analyzed for the singularly perturbed convection-diffusion-reaction problems. The proposed method is based on the idea of replacing the standard gradient (derivative) and convection derivative by modified weak gradient (derivative) and modified weak convection derivative, respectively, over piecewise polynomials of degree $k\geq1$. The present method is parameter-free and has less degree of freedom compared to the weak Galerkin finite element method. Stability and convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy norm are proved. The method is uniformly convergent, i.e., the results hold uniformly regardless of the value of the perturbation parameter. Numerical experiments confirm these theoretical findings on Shishkin meshes. The numerical examples are also carried out on B-S meshes to confirm the theoretical results. Moreover, the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2-$ norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L_\infty-$ norm.

Linearization of the finite element method for gradient flows by Newton’s method

2020 ◽  
Author(s):  
Georgios Akrivis ◽  
Buyang Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Solutions ◽  
Resolvent Estimate ◽  
Optimal Order ◽  
Gradient Flows ◽  
The Finite Element Method ◽  
Element Method

Abstract The implicit Euler scheme for nonlinear partial differential equations of gradient flows is linearized by Newton’s method, discretized in space by the finite element method. With two Newton iterations at each time level, almost optimal order convergence of the numerical solutions is established in both the $L^q(\varOmega )$ and $W^{1,q}(\varOmega )$ norms. The proof is based on techniques utilizing the resolvent estimate of elliptic operators on $L^q(\varOmega )$ and the maximal $L^p$-regularity of fully discrete finite element solutions on $W^{-1,q}(\varOmega )$.

scholarly journals A cut finite element method for a model of pressure in fractured media

2020 ◽  
Vol 146 (4) ◽  
pp. 783-818
Author(s):  
Erik Burman ◽  
Peter Hansbo ◽  
Mats G. Larson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Pressure Field ◽  
A Priori ◽  
Full Range ◽  
Fractured Media ◽  
Optimal Order ◽  
A Priori Error Estimate ◽  
Background Mesh ◽  
Element Method

AbstractWe develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.

scholarly journals L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems

2019 ◽  
Vol 27 (2) ◽  
pp. 85-99
Author(s):  
Christoph Lehrenfeld ◽  
Arnold Reusken
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Analysis ◽  
Error Bounds ◽  
Order Approximation ◽  
High Order ◽  
Optimal Order ◽  
Interface Problem ◽  
High Order Approximation ◽  
Element Method

AbstractIn the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken,IMA J. Numer. Anal.38(2018), No. 3, 1351–1387] ana priorierror analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in theH1-norm. In this paper we extend this analysis and derive optimalL2-error bounds.

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