Monotone and conservative difference schemes for elliptic equations with mixed derivatives¹

The author thanks Prof. Oleg Iliev and Prof. Raimondas Čiegis for the statement of the problem, Prof. Piotr Matus and Dr. Mikhail Chuiko for the discussion and useful comments

Abstract

In the paper elliptic equations with alternating‐sign coefficients at mixed derivatives are considered. For such equations new difference schemes of the second order of approximation are developed. The proposed schemes are conservative and monotone. The constructed algorithms satisfy the grid maximum principle not only for coefficients of constant signs but also for alternating‐sign coefficients at mixed derivatives. The a priori estimates of stability and convergence in the grid norm C are obtained.

Straipsnyje nagrinejamos eliptinio tipo lygtys su mišriomis išvestinemis. Šioms diferencialinems lygtims pasiūlytos naujos antros eiles baigtiniu skirtumu schemos, kurios yra monotoniškos ir konservatyvios. Sukonstruoti algoritmai tenkina skaitini maksimumo principa, kai koeficientai prie mišriuju išvestiniu gali būti bet kokio ženklo. Gauti aprioriniai iverčiai maksimumo normoje. Irodyta baigtiniu skirtumu schemu stabilumas ir konvergavimas.

Key words:

• monotone difference scheme
• conservative difference scheme
• elliptic equations
• mixed derivatives
• grid maximum principle

Notes

The author thanks Prof. Oleg Iliev and Prof. Raimondas Čiegis for the statement of the problem, Prof. Piotr Matus and Dr. Mikhail Chuiko for the discussion and useful comments