scholarly journals Discontinuous Mixed Covolume Methods for Parabolic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ailing Zhu ◽  
Ziwen Jiang
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Error Estimates ◽  
Parabolic Problems ◽  
Optimal Order ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Optimal Order Error Estimates ◽  
Backward Euler

We present the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume schemes for parabolic problems on triangular meshes. We give the error analysis of the discontinuous mixed covolume schemes and obtain optimal order error estimates in discontinuousHdivand first-order error estimate inL2.

scholarly journals Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ailing Zhu
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Optimal Order ◽  
Triangular Meshes ◽  
Element Space ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Mixed Element

The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuousH(div)and first-order error estimate inL2are obtained with the lowest order Raviart-Thomas mixed element space.

Long-time L∞(L2) a posteriori error estimates for fully discrete parabolic problems

2018 ◽  
Vol 40 (1) ◽  
pp. 498-529 ◽  
Author(s):  
Oliver J Sutton
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Parabolic Problems ◽  
Optimal Order ◽  
Continuous Family ◽  
Order Error ◽  
Backward Euler ◽  
Long Time

Abstract Computable estimates for the error of finite element discretisations of parabolic problems in the $L^{\infty }(0,T; L^2(\varOmega ))$ norm are developed, which exhibit constant effectivities (the ratio of the estimated error to the true error) with respect to the simulation time. These estimates, which are of optimal order, represent a significant advantage for long-time simulations and are derived using energy techniques based on elliptic reconstructions. The effectivities of previous optimal-order error estimates in this norm derived using energy techniques are shown numerically to grow in proportion to either the simulation duration or its square root, a key disadvantage compared with earlier estimators derived using parabolic duality arguments. The new estimates form a continuous family, almost all of which are new, reproducing certain familiar energy-based estimates well suited for short-time simulations and not available through the parabolic duality framework. For clarity, we demonstrate the technique applied to a linear parabolic problem discretised using standard conforming finite element methods in space coupled with backward Euler and Crank–Nicolson time discretisations, although it can be applied much more widely.

scholarly journals Analysis of finite element methods for vector Laplacians on surfaces

2019 ◽  
Vol 40 (3) ◽  
pp. 1652-1701 ◽  
Author(s):  
Peter Hansbo ◽  
Mats G Larson ◽  
Karl Larsson
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Vector Fields ◽  
Optimal Order ◽  
Order Error ◽  
Optimal Order Error Estimates ◽  
Elastic Membranes ◽  
Tangential Vector ◽  
Higher Order Finite Elements ◽  
Lagrange Elements

Abstract We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in ${\mathbb{R}}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a ${\mathbb{R}}^3$ vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and $L^2$ norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.

scholarly journals The Backward Euler Fully Discrete Finite Volume Method for the Problem of Purely Longitudinal Motion of a Homogeneous Bar

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Ziwen Jiang ◽  
Deren Xie
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Optimal Order ◽  
Longitudinal Motion ◽  
Order Error ◽  
Fully Discrete ◽  
Backward Euler ◽  
Volume Method

We present a linear backward Euler fully discrete finite volume method for the initial-boundary-value problem of purely longitudinal motion of a homogeneous bar and an give optimal order error estimates inL2andH1norms. Furthermore, we obtain the superconvergence error estimate of the generalized projection of the solutionuinH1norm. Numerical experiment illustrates the convergence and stability of this scheme.

scholarly journals Optimal Order Error Estimates of a Modified Nonconforming Rotated Q1 IFEM for Interface Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Pei Yin ◽  
Hongyun Yue ◽  
Hongbo Guan
Keyword(s):  
Error Estimates ◽  
State Equation ◽  
Interface Problems ◽  
Optimal Order ◽  
Order Error ◽  
Elliptic Interface Problems ◽  
Immersed Finite Element ◽  
Optimal Order Error Estimates ◽  
Variational Discretization ◽  
Theoretical Results

This paper presents a new numerical method and analysis for solving second-order elliptic interface problems. The method uses a modified nonconforming rotated Q1 immersed finite element (IFE) space to discretize the state equation required in the variational discretization approach. Optimal order error estimates are derived in L2-norm and broken energy norm. Numerical examples are provided to confirm the theoretical results.

scholarly journals Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations

2020 ◽  
Author(s):  
Norikazu Saito
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Parabolic Type ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Step Method ◽  
Order Error ◽  
Time Stepping ◽  
Optimal Order Error Estimates

Abstract The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška–Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it differs from previous works by which the method is formulated as the one-step method. We apply our abstract results to the finite element approximation of a second-order parabolic equation with space-time variable coefficient functions in a polyhedral domain, and derive the optimal order error estimates in several norms.

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