The optimization techniques are useful in many areas as demonstrated with our MILAMIN applications for thermal and incompressible flow (Stokes) problems. In order to avoid unnecessary data transfers between RAM and CPU cache we introduce loop blocking. Applying loop interchange allows us to use BLAS on large matrices. Applied optimizations eliminate Basic Linear Algebra Subprograms (BLAS) drawbacks when multiplying small matrices, reduce operation count and memory requirements when dealing with symmetric matrices, and increase data transfer efficiency by maximizing cache reuse. An alternative approach is developed that sustains high performance for any system size. The performance bottlenecks of the standard FEM algorithm are analyzed. One of the main technical focuses of this paper is the optimization of the global matrix computations. While some of the components rely on readily available routines, we develop others from scratch and make sure that all of them work together efficiently. It has been our special aim that all the components of MILAMIN perform efficiently, individually and as a package. In this paper we provide the technical knowledge required to develop such models without the need to buy a commercial FEM package, programming compiler-language code, or hiring a computer specialist. MILAMIN allows the user to achieve numerical resolutions that are necessary to resolve the heterogeneous nature of geological materials. We therefore developed “Million a Minute” (MILAMIN), an efficient MATLAB implementation of FEM that is capable of setting up, solving, and postprocessing two-dimensional problems with one million unknowns in one minute on a modern desktop computer. Practical applications often require high-resolution models that necessitate advanced computational strategies. The finite element method (FEM) combined with unstructured meshes forms an elegant and versatile approach capable of dealing with the complexities of problems in Earth science.

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