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# Finite Element Approximate Inverse Preconditioning for solving 3D Biharmonic Problems on Shared Memory Systems

G.A. Gravvanis1, K.M. Giannoutakis2
Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, University Campus, Kimmeria, GR 67100, Xanthi, Greece
Centre for Research and Technology Hellas, Informatics and Telematics Institute, 6th km Charilaou-Thermi, GR 57001, Thessaloniki, Greece

Computer Modeling in Engineering & Sciences 2011, 71(4), 305-330. https://doi.org/10.3970/cmes.2011.071.305

### Abstract

In this paper we present parallel explicit approximate inverse matrix techniques for solving sparse linear systems on shared memory systems, which are derived using the finite element method for biharmonic equations in three space variables. Our approach for solving such equations is by considering the biharmonic equation as a coupled equation approach (pair of Poisson equation), using a FE approximation scheme, yielding an inner-outer iteration method. Additionally, parallel approximate inverse matrix algorithms are introduced for the efficient solution of sparse linear systems, based on an anti-diagonal computational approach that eliminates the data dependencies. Parallel explicit preconditioned conjugate gradient-type schemes in conjunction with parallel approximate inverse matrix algorithms are presented for the efficient solution of sparse linear systems. Theoretical estimates on computational complexity of the parallel explicit preconditioned conjugate gradient method along with theoretical speedups and efficiency are also presented. Applications of the proposed methods on characteristic biharmonic problems are discussed and numerical results are given.

### Keywords

Biharmonic equations, finite element method, sparse linear systems, approximate factorization procedures, parallel approximate inverse matrix algorithms, parallel preconditioned conjugate gradient methods, shared memory systems.

Gravvanis, G., Giannoutakis, K. (2011). Finite Element Approximate Inverse Preconditioning for solving 3D Biharmonic Problems on Shared Memory Systems. CMES-Computer Modeling in Engineering & Sciences, 71(4), 305–330.

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