In this paper,a two-grid mixed finite element method(MFEM)of implicit Backward Euler(BE)formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the n...In this paper,a two-grid mixed finite element method(MFEM)of implicit Backward Euler(BE)formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the nonconforming EQ_(1)^(rot) element.In this approach,the original nonlinear system is solved on the coarse mesh through the Newton iteration method,and then the linear system is computed on the fine mesh with Taylor’s expansion.Based on the high accuracy results of the chosen element,the uniform superclose and superconvergent estimates in the broken H^(1)-norm are derived,which are independent of the negative powers of the perturbation parameter appeared in the considered problem.Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.展开更多
In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Un...In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator A2, the bi-wave operator □^2 is not an elliptic operator, so the energy space for the bi-wave equation is much larger than the energy space for the biharmonic equation. This then makes it possible to construct low order conforming finite elements for the bi-wave equation. However, the existence and construction of such finite elements strongly depends on the mesh. In the paper, we first characterize mesh conditions which allow and not allow construction of low order conforming finite elements for approximating the bi-wave equation. We then construct a cubic and a quartic conforming finite element. It is proved that both elements have the desired approximation properties, and give optimal order error estimates in the energy norm, suboptimal （and optimal in some cases） order error estimates in the H1 and L^2 norm. Finally, numerical experiments are presented to guage the efficiency of the proposed finite element methods and to validate the theoretical error bounds.展开更多
In clinical practice, dentists sometimes encounter phenomena that cannot be explained by modern western medical concepts;for example, the patient’s medical symptoms improve by bringing medicines or dentures close to ...In clinical practice, dentists sometimes encounter phenomena that cannot be explained by modern western medical concepts;for example, the patient’s medical symptoms improve by bringing medicines or dentures close to the body. Although it seems difficult to completely elucidate the mechanism through modern western medicine, it can be explained using quantum mechanics. The quantum, the smallest unit of matter composition, exhibits wave-particle duality. The fact that symptoms can be improved simply by bringing dentures or medicines closer to the body indicates that the waves emitted by dentures or medicines interfere with the pathological waves emitted by the pathological site. Thus, the pathological waves are deformed and lead to a change in symptoms. In this way, quantum theory can explain phenomena that are difficult to elucidate in conventional medicine, which are encountered in clinical practice. So far, the author has presented a case of difficulty in raising the upper limb where the symptoms improved without the need for dentures in the mouth by adjusting the dentures outside the mouth. This time, the author would like to introduce a case which the patient’s knee pain improved by adjusting the dentures outside the mouth.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12201640,12071443).
文摘In this paper,a two-grid mixed finite element method(MFEM)of implicit Backward Euler(BE)formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the nonconforming EQ_(1)^(rot) element.In this approach,the original nonlinear system is solved on the coarse mesh through the Newton iteration method,and then the linear system is computed on the fine mesh with Taylor’s expansion.Based on the high accuracy results of the chosen element,the uniform superclose and superconvergent estimates in the broken H^(1)-norm are derived,which are independent of the negative powers of the perturbation parameter appeared in the considered problem.Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.
基金partially supported by the NSF grant DMS-0710831
文摘In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator A2, the bi-wave operator □^2 is not an elliptic operator, so the energy space for the bi-wave equation is much larger than the energy space for the biharmonic equation. This then makes it possible to construct low order conforming finite elements for the bi-wave equation. However, the existence and construction of such finite elements strongly depends on the mesh. In the paper, we first characterize mesh conditions which allow and not allow construction of low order conforming finite elements for approximating the bi-wave equation. We then construct a cubic and a quartic conforming finite element. It is proved that both elements have the desired approximation properties, and give optimal order error estimates in the energy norm, suboptimal （and optimal in some cases） order error estimates in the H1 and L^2 norm. Finally, numerical experiments are presented to guage the efficiency of the proposed finite element methods and to validate the theoretical error bounds.
文摘In clinical practice, dentists sometimes encounter phenomena that cannot be explained by modern western medical concepts;for example, the patient’s medical symptoms improve by bringing medicines or dentures close to the body. Although it seems difficult to completely elucidate the mechanism through modern western medicine, it can be explained using quantum mechanics. The quantum, the smallest unit of matter composition, exhibits wave-particle duality. The fact that symptoms can be improved simply by bringing dentures or medicines closer to the body indicates that the waves emitted by dentures or medicines interfere with the pathological waves emitted by the pathological site. Thus, the pathological waves are deformed and lead to a change in symptoms. In this way, quantum theory can explain phenomena that are difficult to elucidate in conventional medicine, which are encountered in clinical practice. So far, the author has presented a case of difficulty in raising the upper limb where the symptoms improved without the need for dentures in the mouth by adjusting the dentures outside the mouth. This time, the author would like to introduce a case which the patient’s knee pain improved by adjusting the dentures outside the mouth.
