Numerical approximation of nonlinear SPDE’s
Martin Ondreját, Andreas Prohl, Noel J. Walkington
Abstract
Abstract The numerical analysis of stochastic parabolic partial differential equations of the form $$\begin{aligned} du + A(u)\, dt = f \,dt + g \, dW, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>W</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> is surveyed, where A is a nonlinear partial operator and W a Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory.