Is the use of "this" here incorrect grammatically?

If you are referring to the whole thesis that will be presented for your degree, it is already clear to the reader that your Galerkin solver is part of the thesis. Why say it again? "A discontinuous Galerkin solver was developed." is sufficient.

I add that it used to be conventional to use the passive voice "...was developed" but that you might also use the (preferable) active voice "I developed a discontinuous Galerkin solver".

If on the other hand the context is that you have recently introduced a sub-thesis of some sort, "... of this thesis" would be acceptable, because this refers to what you recently mentioned.

For example: "One thesis (or proposition?) is that, just as they have been used on the sphere and ellipsoid, DGMs may applied to the shallow water equations on a hyperboloid. A discontinuous Galerkin solver was developed as part of this thesis."