Publicación: A PRIORI AND A POSTERIORI ERROR ESTIMATES FOR A VIRTUAL ELEMENT SPECTRAL ANALYSIS FOR THE ELASTICITY EQUATIONS
Fecha
2020
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IMA JOURNAL OF NUMERICAL ANALYSIS
Resumen
WE PRESENT A PRIORI AND A POSTERIORI ERROR ANALYSIS OF A VIRTUAL ELEMENT METHOD (VEM) TO APPROXIMATE THE VIBRATION FREQUENCIES AND MODES OF AN ELASTIC SOLID. WE ANALYZE A VARIATIONAL FORMULATION RELYING ONLY ON THE SOLID DISPLACEMENT AND PROPOSE AN $H^1({\OMEGA})$-CONFORMING DISCRETIZATION BY MEANS OF VEM. UNDER STANDARD ASSUMPTIONS ON THE COMPUTATIONAL DOMAIN, WE SHOW THAT THE RESULTING SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE AN OPTIMAL ORDER ERROR ESTIMATE FOR THE EIGENFUNCTIONS AND A DOUBLE ORDER FOR THE EIGENVALUES. SINCE, THE VEM HAS THE ADVANTAGE OF USING GENERAL POLYGONAL MESHES, WHICH ALLOWS IMPLEMENTING EFFICIENTLY MESH REFINEMENT STRATEGIES, WE ALSO INTRODUCE A RESIDUAL-TYPE A POSTERIORI ERROR ESTIMATOR AND PROVE ITS RELIABILITY AND EFFICIENCY. WE USE THE CORRESPONDING ERROR ESTIMATOR TO DRIVE AN ADAPTIVE SCHEME. FINALLY, WE REPORT THE RESULTS OF A COUPLE OF NUMERICAL TESTS THAT ALLOW US TO ASSESS THE PERFORMANCE OF THIS APPROACH.
