scholarly journals Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations

2018 ◽  
Vol 52 (6) ◽  
pp. 2307-2325 ◽  
Author(s):  
Dominik Meidner ◽  
Boris Vexler
Keyword(s):  
Error Estimates ◽  
Uniform Boundedness ◽  
Growth Conditions ◽  
Optimal Error Estimates ◽  
Semilinear Parabolic Equation ◽  
Optimal Error ◽  
Implicit Euler Scheme ◽  
Fully Discrete ◽  
Galerkin Approximations ◽  
Semilinear Parabolic

We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme) and with conforming finite elements in space. The main contribution of this paper is the proof of the uniform boundedness of the discrete solution. This allows us to obtain optimal error estimates with respect to various norms.

Optimal Error Estimates for a Fully Discrete Euler Scheme for Decoupled Forward Backward Stochastic Differential Equations

2017 ◽  
Vol 7 (3) ◽  
pp. 548-565
Author(s):  
Bo Gong ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Error Estimates ◽  
Backward Stochastic Differential Equations ◽  
Optimal Error Estimates ◽  
Euler Scheme ◽  
Optimal Error ◽  
Fully Discrete ◽  
Numerical Approaches

AbstractIn error estimates of various numerical approaches for solving decoupled forward backward stochastic differential equations (FBSDEs), the rate of convergence for one variable is usually less than for the other. Under slightly strengthened smoothness assumptions, we show that the fully discrete Euler scheme admits a first-order rate of convergence for both variables.

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