For the transient behavior of a semiconductor device, the modified method of characteristics with alternating-direction finite element procedures for nonrectangular region is put forward. Some techniques, such as calc...For the transient behavior of a semiconductor device, the modified method of characteristics with alternating-direction finite element procedures for nonrectangular region is put forward. Some techniques, such as calculus of variations, isoparametric transformation，patch approximation, operator-splitting, characteristic method, symmetrical reflection,energy method, negative norm estimate and a prior estimates and techniques, are employed. In the nonrectangular region case, optimal order estimates in L^2 norm are derived for the error in the approximation solution. Thus the well-known theoretical problem has been thoroughly and completely solved.展开更多
The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentr...The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin method.We estimate the error of the numerical solutions in the sense of the Lqnorm.To linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton iteration.The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid.Error estimation for the two-grid solutions is analyzed in detail.It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.展开更多
The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not s...The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not small.They also extend them to the weak solutions to parabolic equations.展开更多
In this paper, we shall introduce the concept of the Bessel （Riesz） potential Kothe function spaces X<sup>s</sup> （<sup>s</sup>） and give some dual estimates for a class of operators determ...In this paper, we shall introduce the concept of the Bessel （Riesz） potential Kothe function spaces X<sup>s</sup> （<sup>s</sup>） and give some dual estimates for a class of operators determined by a semi-group in the spaces L<sup>q</sup> （-T, T; X<sup>s</sup>） （L<sup>q</sup>（-T, T; <sup>s</sup> ））. Moreover, some time-space L<sup>P</sup>-L<sup>P</sup><sup> </sup>estimates for the semi-group exp(it（-△）<sup>m/2</sup>) and the operator A:=∫<sub>0</sub><sup>t</sup> exp(i（t-τ）（-△）<sup>m/2</sup>). dτ in the Lebesgue-Besov spaces L<sup>q</sup>(-T, T; <sub>p,2</sub><sup>S</sup>) are given. On the basis of these results, in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.展开更多
Characteristic finite difference fractional step schemes are put forward. The electric potential equation is described by a seven-point finite difference scheme, and the electron and hole concentration equations are t...Characteristic finite difference fractional step schemes are put forward. The electric potential equation is described by a seven-point finite difference scheme, and the electron and hole concentration equations are treated by a kind of characteristic finite difference fractional step methods. The temperature equation is described by a fractional step method. Thick and thin grids are made use of to form a complete set. Piecewise threefold quadratic interpolation, symmetrical extension, calculus of variations, commutativity of operator product, decomposition of high order difference operators and prior estimates are also made use of. Optimal order estimates in l2 norm are derived to determine the error of the approximate solution. The well-known problem is thorongley and completely solred.展开更多
基金This research is supported by the Major State Basic Research Program of China (Grant No. 19990328), the National Tackling Key Problem Program, the National Science Foundation of China (Grant Nos. 10271066 and 10372052), the Doctorate Foundation of th
文摘For the transient behavior of a semiconductor device, the modified method of characteristics with alternating-direction finite element procedures for nonrectangular region is put forward. Some techniques, such as calculus of variations, isoparametric transformation，patch approximation, operator-splitting, characteristic method, symmetrical reflection,energy method, negative norm estimate and a prior estimates and techniques, are employed. In the nonrectangular region case, optimal order estimates in L^2 norm are derived for the error in the approximation solution. Thus the well-known theoretical problem has been thoroughly and completely solved.
基金Project supported by the State Key Program of National Natural Science Foundation of China(No.11931003)the National Natural Science Foundation of China(Nos.41974133,11671157,11971410)。
文摘The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin method.We estimate the error of the numerical solutions in the sense of the Lqnorm.To linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton iteration.The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid.Error estimation for the two-grid solutions is analyzed in detail.It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.
基金supported by the National Key R&D Program of China(No.2021YFA1003001).
文摘The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not small.They also extend them to the weak solutions to parabolic equations.
基金Supported in part by the Doctoral Research Foundation of Hebei Province
文摘In this paper, we shall introduce the concept of the Bessel （Riesz） potential Kothe function spaces X<sup>s</sup> （<sup>s</sup>） and give some dual estimates for a class of operators determined by a semi-group in the spaces L<sup>q</sup> （-T, T; X<sup>s</sup>） （L<sup>q</sup>（-T, T; <sup>s</sup> ））. Moreover, some time-space L<sup>P</sup>-L<sup>P</sup><sup> </sup>estimates for the semi-group exp(it（-△）<sup>m/2</sup>) and the operator A:=∫<sub>0</sub><sup>t</sup> exp(i（t-τ）（-△）<sup>m/2</sup>). dτ in the Lebesgue-Besov spaces L<sup>q</sup>(-T, T; <sub>p,2</sub><sup>S</sup>) are given. On the basis of these results, in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.
基金This work is supported by the Major State Basic Research Program of China (19990328), the National Tackling Key Problem Program, the National Science Foundation of China (10271066 and 0372052), and the Doctorate Foundation of the Ministry of Education of China (20030422047).
文摘Characteristic finite difference fractional step schemes are put forward. The electric potential equation is described by a seven-point finite difference scheme, and the electron and hole concentration equations are treated by a kind of characteristic finite difference fractional step methods. The temperature equation is described by a fractional step method. Thick and thin grids are made use of to form a complete set. Piecewise threefold quadratic interpolation, symmetrical extension, calculus of variations, commutativity of operator product, decomposition of high order difference operators and prior estimates are also made use of. Optimal order estimates in l2 norm are derived to determine the error of the approximate solution. The well-known problem is thorongley and completely solred.
