A high performance algorithm for the implementation of Arbitrary Lagrangian and Eulerian (ALE) moving mesh scheme for both 2D and 3D Fluid Structure Interaction (FSI) problems for the shared and distributed memory systems is discussed in the the- sis. OpenMP library is used to implement parallel programs on shared memory systems whereas message passing interface (MPI) is employed to write parallel programs on dis- tributed memory systems. Moving mesh techniques are the integral part of a wider class of fluid mechanics problems that involve moving and deforming spatial domains, namely, free-surface flows and FSI. The moving mesh technique adopted in this work is based on the notion of nodes relocation, subjected to certain evolution as well as constraint conditions. A conjugate gradient method augmented with a preconditioning is employed for the solution of the resulting system of equations. The proposed algorithm, firstly, re- orders and partitions the mesh using an efficient divide and conquer approach and then parallelizes the ALE moving mesh. Different mesh partitioning algorithms are discussed, which include the octree method, and k-way graph partitioning technique using Parmetis library. Numerical simulations are conducted on AMD Opteron and Intel Xeon processors, and unstructured triangular and tetrahedral meshes are used for the 2D and 3D prob- lems. The better results, in terms of the speedup, are obtained for the shared memory system than the distributed memory system for both the 2D and 3D problems. The quality of meshes is checked by comparing the element Jacobians in the reference and current meshes, and by keeping track of the change in the interior angles in triangles and tetrahedrons. The proposed parallel mesh reordering algorithm using sampling approach for work load re-distribution concluded 51% of average efficiency in term of the speedup for shared memory systems. The overall maximum speedup of 6.37, for the shared mem- ory system, is achieved using eight processing elements (PEs) as compared to 4.11 for the distributed memory system including twelve PEs. As a case study, the thesis also discusses the high performance implementation of a stabilized mixed finite element method for Darcy flow using MPI library. It has a lot of practical applications in the field of petroleum engineering and earth sciences especially, where the flow of fluid is of interest in a permeable porous medium. The maximum speedup of 12.24 is achieved using 28 PEs by incorporating the proposed mesh partition- ing algorithm. Outline Chapter 1 defines and introduces the problem statement and Chapter 2 gives the gen- eral introduction of the thesis. Chapter 3 presents the literature review of ALE moving mesh generation, stabilized mixed finite element methods, k-way graph partitioning algo- rithm and tree based spatial data structures. Chapter 4 mathematically formulates the ALE mesh generation problem and presents the serial algorithm for optimization using the preconditioned conjugate gradient method. Chapter 5 presents a mesh reordering algorithm based on quadtree/octree and quick sort techniques. Chapter 6 discusses the parallelization part of mesh reordering algorithm based on a sampling approach and also discusses the experimental results for the shared memory systems. Detailed discussion about the mesh partitioning and experimental results using MPI are given in Chapter 7. Chapter 8 briefly describes the stabilized finite element method for Darcy Flow and dis- cusses the results of 2D problems for a distributed memory system. Finally, conclusions are drawn in Chapter 9 and future work is presented in Chapter 10.