The paper presents the graph grammar based multi-thread multi-frontal parallel direct solver for one and two dimensional Finite Difference Method (FDM). The multi-frontal solver algorithm has been expressed by graph grammar productions. Each production represents an atomic task that internally must be executed in serial. The sequence of graph grammar productions modeling the execution of the solver has been associated with the alphabet for the trace theory analysis. The dependency relation between tasks has been introduced based on the analysis of the solver algorithm. The sequence of productions has been transformed into the Foata Normal Form (FNF). The parallel solver algorithm has been implemented and tested on NVIDIA Cuda multi-core graphic card. The tasks have been scheduled according to the classes in the FNF.