Maximum Norm Regularity of Implicit Difference Methods for Parabolic Equations

Abstract

<p>We prove maximum norm regularity properties of L-stable finite difference</p><p>methods for linear-second order parabolic equations with coefficients</p><p>independent of time, valid for large time steps. These results are almost</p><p>sharp; the regularity property for first differences of the numerical solution</p><p>is of the same form as that of the continuous problem, and the regularity</p><p>property for second differences is the same as the continuous problem except for</p><p>logarithmic factors. </p><p>This generalizes a result proved by Beale valid for the constant-coefficient</p><p>diffusion equation, and is in the spirit of work by Aronson, Widlund and</p><p>Thome&eacute.</p><p>To prove maximum norm regularity properties for the homogeneous problem, </p><p> we introduce a semi-discrete problem (discrete in space, continuous in time).</p><p>We estimate the semi-discrete evolution operator and its spatial differences on</p><p>a sector of the complex plan by constructing a fundamental solution.</p><p>The semi-discrete fundamental solution is obtained from the fundamental solution to the frozen coefficient problem by adding a correction term found through an iterative process.</p><p>From the bounds obtained on the evolution operator and its spatial differences,</p><p>we find bounds</p><p>on the resolvent of the discrete elliptic operator and its differences through</p><p>the Laplace transform</p><p>representation of the resolvent. Using the resolvent estimates and the</p><p>assumed stability properties of the time-stepping method in the Cauchy integral</p><p>representation of the fully discrete solution operator</p><p>yields the homogeneous regularity result.</p><p>Maximum norm regularity results for the inhomogeneous</p><p>problem follow from the homogeneous results using Duhamel's principle. The results for the inhomogeneous</p><p>problem</p><p>imply that when the time step is taken proportional to the grid width, the rate of convergence of the numerical solution and its first</p><p>differences is second-order in space, and the rate of convergence for second</p><p>differences</p><p>is second-order except for logarithmic factors .</p><p>As an application of the theory, we prove almost sharp maximum norm resolvent estimates for divergence</p><p>form elliptic operators on spatially periodic grid functions. Such operators are invertible, with inverses and their first differences bounded in maximum norm, uniformly in the grid width. Second differences of the inverse operator are bounded except for logarithmic factors.</p>