| 1. |
- Beilina, Larisa, 1970, et al.
(författare)
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An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations
- 2016
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 61:3, s. 253-286
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Tidskriftsartikel (refereegranskat)abstract
- We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate the corresponding adaptive algorithm. Our numerical experiments justify the efficiency of our a posteriori estimates and show significant improvement of the reconstructions obtained on locally adaptively refined meshes.
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| 2. |
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| 3. |
- Beilina, Larisa, 1970, et al.
(författare)
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On the Maxwell-wave equation coupling problem and its explicit finite-element solution
- 2023
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Ingår i: Applications of Mathematics. - 1572-9109 .- 0862-7940. ; 68:1, s. 75-98
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Tidskriftsartikel (refereegranskat)abstract
- It is well known that in the case of constant dielectric permittivity and magnetic permeability, the electric field solving the Maxwell’s equations is also a solution to the wave equation. The converse is also true under certain conditions. Here we study an intermediate situation in which the magnetic permeability is constant and a region with variable dielectric permittivity is surrounded by a region with a constant one, in which the unknown field satisfies the wave equation. In this case, such a field will be the solution of Maxwell’s equation in the whole domain, as long as proper conditions are prescribed on its boundary. We show that an explicit finite-element scheme can be used to solve the resulting Maxwell-wave equation coupling problem in an inexpensive and reliable way. Optimal convergence in natural norms under reasonable assumptions holds for such a scheme, which is certified by numerical exemplification.
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| 4. |
- Byström, Johan, et al.
(författare)
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Bounds and numerical results for the homogenized degenerated p-Poisson equation
- 2004
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Ingår i: Applications of Mathematics. - 0862-7940 .- 1572-9109. ; 49:2, s. 111-122
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.
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| 5. |
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| 6. |
- Danielsson, Tatiana, et al.
(författare)
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Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales
- 2024
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 69:1, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems. At the end of the paper an example is given to concretize the use of the main result.
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| 7. |
- Essel, Emmanuel Kwame, et al.
(författare)
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Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence
- 2010
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 55:4, s. 305-327
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Tidskriftsartikel (refereegranskat)abstract
- We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe's method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.
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| 8. |
- Flodén, Liselott, et al.
(författare)
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Homogenization of some parabolic operators with several time scales
- 2007
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Ingår i: Applications of Mathematics. - 0862-7940 .- 1572-9109. ; 52:5, s. 431-446
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Tidskriftsartikel (refereegranskat)abstract
- The main focus in this paper is on homogenization of a parabolic problem with two local time scales. Under certain assumptions on the coefficient a, there exists a G-limit b, which we characterize by means of multiscale techniques for r>0, r≠1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.
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| 9. |
- Francu, J., et al.
(författare)
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Some remarks on two-scale convergence and periodic unfolding
- 2012
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 57:4, s. 359-375
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Tidskriftsartikel (refereegranskat)abstract
- The paper discusses some aspects of the adjoint definition of two-scale convergence based on periodic unfolding. As is known this approach removes problems concerning choice of the appropriate space for admissible test functions. The paper proposes a modified unfolding which conserves integral of the unfolded function and hence simplifies the proofs and its application in homogenization theory. The article provides also a self-contained introduction to two-scale convergence and gives ideas for generalization to non-periodic homogenization.
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| 10. |
- Holmbom, Anders
(författare)
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Homogenization of parabolic equations an alternative approach and some corrector-type results
- 1997
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Ingår i: Applications of Mathematics. - 0862-7940 .- 1572-9109. ; 42:5, s. 321-343
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Tidskriftsartikel (refereegranskat)abstract
- We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type results and benefit from some interpolation properties of Sobolev spaces to identify regularity assumptions strong enough for such results to hold. partial differential equations - homogenization - two-scale convergence - linear parabolic equations - oscillating coefficients in space and time variable - dissimilar speeds of oscillation - admissible test functions - corrector results - compactness result - interpolationThis research was supported by The Swedish Research Council for the Engineering Sciences (TFR), The Swedish National Board for Industrial and Technological Development (NUTEK), and The Country of Jämtland Research Foundation
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