We present a PDE solver based  on the multisplitting-Newton approach using a GPU
accelerated  sparse linear solver  as inner  core.  The  multisplitting approach
allows  us to  make use  of asynchronism  which sensibly  decreases  the overall
computation  time  by performing  an  implicit  overlapping  of computations  by
communications. Moreover, as  most PDE problems come from  physical modelling in
which the  dependency scheme produces sparse  matrices, we propose  as our inner
linear solver, a sparse one, designed to work with structured matrices where all
the non-zeros are on a few  diagonals. Finally, several benchmarks point out the
interest of using asynchronous  algorithms together with local accelerators like
GPUs.