Fast solvers for mesh-based computations

Content: Multi-Frontal Direct Solver Algorithm for Tri-Diagonal and Block-Diagonal One-Dimensional Problems Derivation of the Linear System for One-Dimensional Finite Difference MethodAlgebraic Algorithm of the Multi-Frontal Solver Graph-Grammar Based Model of Concurrency of the Multi-Frontal Solver AlgorithmOne-Dimensional Finite Element Method with Linear Basis FunctionsOne-Dimensional Isogeometric Collocation Method with Quadratic B-Splines One-Dimensional Finite Element Method with Buble Basis Functions One-Dimensional Non-Stationary Problems Euler Scheme with Respect to Time Mixed with Finite Element Method with LinearBasis Functions with Respect to Space α-Scheme with Respect to Time Mixed with Method with Linear Basis Functionsfor SpaceMulti-Frontal Direct Solver Algorithm for Multi-Diagonal One-Dimensional Problems One-Dimensional Collocation Method with Higher Order B-Splines One-Dimensional Isogeometric Finite Element MethodMulti-Frontal Direct Solver Algorithm for Two-Dimensional Grids with Block DiagonalStructure of the Matrix Two-Dimensional Projection Problem with Linear Basis Functions Two-Dimensional Mesh with Anisotropic Edge Singularity Two-Dimensional Mesh with Point Singularity Multi-Frontal Direct Solver Algorithm for Three-Dimensional Grids with Block Diagonal Structure of the Matrix Three-Dimensional Projection Problem with Linear Basis Functions Three-Dimensional Mesh with Anisotropic Face SingularityThree-Dimensional Mesh with Anisotropic Edge Singularity Three-Dimensional Mesh with Point SingularityMulti-Frontal Direct Solver Algorithm for Two-Dimensional Isogeometric FiniteElement MethodIsogeometric Finite Element Method for Two-Dimensional ProblemsGraph-Grammar for Generation of the Elimination TreeGraph-Grammar Productions for the Solver AlgorithmExpressing Partial LU Factorization by BLAS CallsLU Factorization of A(1,1)Multiplication of A(1,2) by the Inverse of A(1,1) Multiplication of b(1) by the Inverse of A(1,1)Matrix Multiplication and Subtraction A(2,2)=A(2,2)-A(2,1)A(1,2) Matrix Vector Multiplication and Subtraction b(2)=b,2)-A(2,1)b(1)ExampleMulti-Frontal Solver Algorithm for Arbitrary Mesh-Based Computations Multi-Frontal Solver Algorithm for Arbitrary GridsHypermatrix Module Elimination Tree ModuleSupernodes System Module Interface Structure of Matrices for Different Two-Dimensional MethodsElimination Trees Elimination Trees and Multi-Frontal Solvers Quasi-Optimal Elimination Tree for Two-Dimensional Mesh with Point SingularityQuasi-Optimal Elimination Tree for Two-Dimensional Mesh with Edge SingularityNested-Dissection Elimination Tree for Two-Dimensional Mesh with Edge SingularityMinimum Degree Tree for Two-Dimensional Mesh with Edge Singularity Estimation of the Number of Floating Point Operations and Memory Usage Elimination Trees for Three-Dimensional GridsReutilization and Reuse of Partial LU Factorizatons Idea of the Reutilization AlgorithmExemplary Implementation of the Reutilization AlgorithmIdea of the Reuse AlgorithmExemplary Implementation of the Reuse AlgorithmNumerical ExperimentsMeasuring the Solver Performance by Means of Execution TimeMeasuring the Solver Performance by Means of the Number of Floating Point Operations (FLOPs)Measuring the Solver Performance by Means of the Efficiency and SpeedupGraph-Grammar Based Multi-Thread GALOIS Solver for Two-Dimensional Grids with SingularitiesGraph-Grammar Based Multi-Thread GALOIS Solver for Three-Dimensional Grids with SingularitiesGraph-Grammar Based GPU Solver for One-Dimensional Isogoemetric Finite Element Method Graph-Grammar Based GPU Solver for Two-Dimensional Isogoemetric Finite Element MethodGraph-Grammar Based Solver for Two-Dimensional Adaptive Finite Element MethodGraph-Grammar Based Solver for Three-Dimensional Adaptive Finite ElementMethod