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To exploit the full potential of GPGPUs for general purpose computing, DOACR parallelism abundant in scientific and engineering applications must be harnessed. However, the presence of cross-iteration data dependences in DOACR loops poses an obstacle to execute their computations concurrently using a massive number of fine-grained threads. This work focuses on iterative PDE solvers rich in DOACR parallelism to identify optimization principles and strategies that allow their efficient mapping to GPGPUs. Our main finding is that certain DOACR loops can be accelerated further on GPGPUs if they are algorithmically restructured (by a domain expert) to be more amendable to GPGPU parallelization, judiciously optimized (by the compiler), and carefully tuned by a performance-tuning tool. We substantiate this finding with a case study by presenting a new parallel SSOR method that admits more efficient data-parallel SIMD execution than red-black SOR on GPGPUs. Our solution is obtained non-conventionally, by starting from a K-layer SSOR method and then parallelizing it by applying a non-dependence-preserving scheme consisting of a new domain decomposition technique followed by a generalized loop tiling. Despite its relatively slower convergence, our new method outperforms red-black SOR by making a better balance between data reuse and parallelism and by trading off convergence rate for SIMD parallelism. Our experimental results highlight the importance of synergy between domain experts, compiler optimizations and performance tuning in maximizing the performance of applications, particularly PDE-based DOACR loops, on GPGPUs.