arXiv subjects
Publications and source records attributed to István Gyöngy.
We give sufficient conditions under which solutions of finite-difference schemes in the space variable for second order possibly degenerate parabolic and elliptic equations admit estimates of spatial derivatives up to any given order independent of the mesh size.
We approximate the solution $u$ of the Cauchy problem $$ \frac{\partial}{\partial t} u(t,x)=Lu(t,x)+f(t,x), \quad (t,x)\in(0,T]\times\bR^d, $$ $$ u(0,x)=u_0(x),\quad x\in\bR^d $$ by splitting the equation into the system $$ \frac{\partial}{\partial t} v_r(t,x)=L_rv_r(t,x)+f_r(t,x), \qquad r=1,2,...,d_1, $$ where $L,L_r$ are second order differential operators, $f$, $f_r$ are functions of $t,x$, such that $L=\sum_r L_r$, $f=\sum_r f_r$. Under natural conditions on solvability in the Sobolev spaces $W^m_p$, we show that for any $k>1$ one can approximate the solution $u$ with an error of order $δ^k$, by an appropriate combination of the solutions $v_r$ along a sequence of time discretization, where $δ$ is proportional to the step size of the grid. This result is obtained by using the time change introduced in [7], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of $δ$.