A Two-Level Method for Pressure Projection Stabilized P1 Nonconforming Approximation of the Semi-Linear Elliptic Equations

2016 ◽  
Vol 8 (3) ◽  
pp. 386-398
Author(s):  
Sufang Zhang ◽  
Hongxia Yan ◽  
Hongen Jia
Keyword(s):  
Data Structures ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Linear Elliptic Equation ◽  
Local Pressure ◽  
Level Method ◽  
Support Sets ◽  
Pressure Projection ◽  
Edge Based ◽  
Computational Properties

Abstract.In this paper, we study a new stabilized method based on the local pressure projection to solve the semi-linear elliptic equation. The proposed scheme combines nonconforming finite element pairs NCP1–P1 triangle element and two-level method, which has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, but have more favorable stability and less support sets. Stability analysis and error estimates have been done. Finally, numerical experiments to check estimates are presented.

scholarly journals A carbuncle cure for the Harten-Lax-van Leer contact (HLLC) scheme using a novel velocity-based sensor

2021 ◽  
Author(s):  
U. S. Vevek ◽  
B. Zang ◽  
T. H. New
Keyword(s):  
Data Structures ◽  
Numerical Experiments ◽  
Additional Data ◽  
Shear Layers ◽  
Hybrid Scheme ◽  
Local Pressure ◽  
Numerical Flux ◽  
Shear Sensor ◽  
Non Local

AbstractA hybrid numerical flux scheme is proposed by adapting the carbuncle-free modified Harten-Lax-van Leer contact (HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact (HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability.

scholarly journals A BDDC Preconditioner for the RotatedQ1FEM for Elliptic Problems with Discontinuous Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yaqin Jiang
Keyword(s):  
Finite Element Method ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Mesh Size ◽  
Discontinuous Coefficients ◽  
Schwarz Methods ◽  
Additive Schwarz ◽  
Variational Form ◽  
Second Order Elliptic Equations ◽  
Additive Schwarz Methods

We propose a BDDC preconditioner for the rotatedQ1finite element method for second order elliptic equations with piecewise but discontinuous coefficients. In the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasioptimal convergence behavior; that is, the condition number of the preconditioned problem is independent of the jumps of the coefficients and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.

scholarly journals On the boundary values of the solutions of linear elliptic equations

1983 ◽  
Vol 27 (1) ◽  
pp. 1-30 ◽  
Author(s):  
J. Chabrowski ◽  
H.B. Thompson
Keyword(s):  
Elliptic Equation ◽  
Sobolev Space ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Sufficient Condition ◽  
Boundary Values ◽  
Linear Elliptic Equation

The purpose of this article is to investigate the traces of weak solutions of a linear elliptic equation. In particular, we obtain a sufficient condition for a solution belonging to the Sobolev space to have an L2-trace on the boundar.

scholarly journals A variant of Clark’s theorem and its applications for nonsmooth functionals without the global symmetric condition

2021 ◽  
Vol 11 (1) ◽  
pp. 285-303
Author(s):  
Chen Huang
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Linear Elliptic Equation ◽  
Clark’S Theorem

Abstract We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].

scholarly journals Instance-Optimal Goal-Oriented Adaptivity

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Michael Innerberger ◽  
Dirk Praetorius
Keyword(s):  
Adaptive Algorithm ◽  
Numerical Experiments ◽  
Dual Problem ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Polynomial Degree ◽  
Energy Error ◽  
Instance Optimality ◽  
Edge Based

AbstractWe consider an adaptive finite element method with arbitrary but fixed polynomial degree {p\geq 1}, where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 2016, 1, 33–68], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. More precisely, the goal error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Numerical experiments underline our theoretical findings.

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