It is well known that for the heat equation on a rectangle, the finite difference alternating direction implicit (ADI) method converges with order two. For the first time in the literature, we bound errors of the finite difference ADI method for the heat equation on a convex set for which it is possible to construct a partition consistent with the boundary. Numerical results indicate that the ADI method may also work for some nonconvex sets for which it is possible to construct a partition consistent with the boundary.
{"title":"Convergence analysis of the finite difference ADI scheme for the heat equation on a convex set","authors":"B. Bialecki, Maxsymillian Dryja, R. Fernandes","doi":"10.1090/MCOM/3653","DOIUrl":"https://doi.org/10.1090/MCOM/3653","url":null,"abstract":"It is well known that for the heat equation on a rectangle, the finite difference alternating direction implicit (ADI) method converges with order two. For the first time in the literature, we bound errors of the finite difference ADI method for the heat equation on a convex set for which it is possible to construct a partition consistent with the boundary. Numerical results indicate that the ADI method may also work for some nonconvex sets for which it is possible to construct a partition consistent with the boundary.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80079052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer $N$ may be rigorously and deterministically factored into primes in at most [ Oleft( frac{N^{1/5} log^{16/5} N}{(loglog N)^{3/5}}right) ] bit operations. This improves on the previous best known result by a factor of $(log log N)^{3/5}$.
{"title":"A log-log speedup for exponent one-fifth deterministic integer factorisation","authors":"David Harvey, Markus Hittmeir","doi":"10.1090/mcom/3708","DOIUrl":"https://doi.org/10.1090/mcom/3708","url":null,"abstract":"Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer $N$ may be rigorously and deterministically factored into primes in at most [ Oleft( frac{N^{1/5} log^{16/5} N}{(loglog N)^{3/5}}right) ] bit operations. This improves on the previous best known result by a factor of $(log log N)^{3/5}$.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74591657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.
{"title":"Accuracy controlled data assimilation for parabolic problems","authors":"W. Dahmen, R. Stevenson, Jan Westerdiep","doi":"10.1090/mcom/3680","DOIUrl":"https://doi.org/10.1090/mcom/3680","url":null,"abstract":"This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81648319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical $ varepsilon = 1$ up to long wave limit regime $ varepsilon ll 1 $ on the natural time scale of the PDE $t= mathcal{O}(frac{1}{varepsilon})$. Most notably our new schemes converge with rates at order $tau varepsilon$ over long times $t= frac{1}{varepsilon}$.
{"title":"Time integrators for dispersive equations in the long wave regime","authors":"M. Calvo, F. Rousset, Katharina Schratz","doi":"10.1090/mcom/3745","DOIUrl":"https://doi.org/10.1090/mcom/3745","url":null,"abstract":"We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical $ varepsilon = 1$ up to long wave limit regime $ varepsilon ll 1 $ on the natural time scale of the PDE $t= mathcal{O}(frac{1}{varepsilon})$. Most notably our new schemes converge with rates at order $tau varepsilon$ over long times $t= frac{1}{varepsilon}$.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84837813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We enumerate the 15768 perfect groups of order up to 2·10, up to isomorphism, thus also completing the missing cases in [HP89]. The work supplements the by now wellunderstood computer classifications of solvable groups, illustrating scope and feasibility of the enumeration process for nonsolvable groups. The algorithmic setup for constructing finite groups of a given order, up to isomorphism, has been well-established, both in theory and in practice, for the construction of groups [BEO02, EHH17]. It proceeds inductively, by constructing extensions of known groups of smaller orders and eliminating isomorphic candidates when they arise. Due to limitations in implementations of underlying routines, this however had been done so far mostly for solvable groups. The aim of this paper is to show the feasibility of generalizing this approach to the case of nonsolvable groups. Instrumental in this has been the calculation of 2-cohomology through confluent rewriting systems, generalizing the method [HEO05, §8.7.2] for solvable groups that uses a PC presentation. The construction process is illustrated by revisiting the enumeration of perfect groups that was started in [HP89] and to extend it to order 2 ·10. In total we find 15768 perfect groups, seeded from the 66 nonabelian simple groups of order up to 2 · 10. Compared with [HP89], this newly provides explicit lists of the groups of orders 61440, 86016, 122880, 172032, 245760, 344064, 368640, 491520, 688128, 737280, 983040 that were omitted in their classification of groups of order up to 10. In this range, the calculations also found five groups (in addition to two groups found already in 2005 by Jack Schmidt) that had been overlooked in [HP89]. Besides serving as examples for testing conjectures, such lists of groups are used as seed in algorithms for the calculation of subgroups of a given finite group [Neu60, CCH01, Hul13], or indeed for the construction of all groups of a given order. All calculations were done using the system GAP [GAP20], which also serves as repository of the resulting group data. The program that performed the classification is available at https://github.com/hulpke/perfect and should allow for easy generalization or extension. In addition to the actual classification result, this work also serves as a prototype of enumeration of nonsolvable groups, extending the work of [BEO02] to the nonsolvable case. It illustrates the feasibility range of current implementations of underlying routines for cohomology, extensions, and isomorphism tests, with a number of general-purpose improvements in the system GAP [GAP20] (that will be part of the 4.12 release) by the author having been motivated by this work. Indeed, the fact, that it took over 30 years since the publication of [HP89] to complete the classification of perfect groups up to order one million, indicates the broad infrastructural 1 ar X iv :2 10 4. 10 82 8v 3 [ m at h. G R ] 6 J ul 2 02 1 requirements of such classifications, wi
{"title":"The perfect groups of order up to two million","authors":"A. Hulpke","doi":"10.1090/mcom/3684","DOIUrl":"https://doi.org/10.1090/mcom/3684","url":null,"abstract":"We enumerate the 15768 perfect groups of order up to 2·10, up to isomorphism, thus also completing the missing cases in [HP89]. The work supplements the by now wellunderstood computer classifications of solvable groups, illustrating scope and feasibility of the enumeration process for nonsolvable groups. The algorithmic setup for constructing finite groups of a given order, up to isomorphism, has been well-established, both in theory and in practice, for the construction of groups [BEO02, EHH17]. It proceeds inductively, by constructing extensions of known groups of smaller orders and eliminating isomorphic candidates when they arise. Due to limitations in implementations of underlying routines, this however had been done so far mostly for solvable groups. The aim of this paper is to show the feasibility of generalizing this approach to the case of nonsolvable groups. Instrumental in this has been the calculation of 2-cohomology through confluent rewriting systems, generalizing the method [HEO05, §8.7.2] for solvable groups that uses a PC presentation. The construction process is illustrated by revisiting the enumeration of perfect groups that was started in [HP89] and to extend it to order 2 ·10. In total we find 15768 perfect groups, seeded from the 66 nonabelian simple groups of order up to 2 · 10. Compared with [HP89], this newly provides explicit lists of the groups of orders 61440, 86016, 122880, 172032, 245760, 344064, 368640, 491520, 688128, 737280, 983040 that were omitted in their classification of groups of order up to 10. In this range, the calculations also found five groups (in addition to two groups found already in 2005 by Jack Schmidt) that had been overlooked in [HP89]. Besides serving as examples for testing conjectures, such lists of groups are used as seed in algorithms for the calculation of subgroups of a given finite group [Neu60, CCH01, Hul13], or indeed for the construction of all groups of a given order. All calculations were done using the system GAP [GAP20], which also serves as repository of the resulting group data. The program that performed the classification is available at https://github.com/hulpke/perfect and should allow for easy generalization or extension. In addition to the actual classification result, this work also serves as a prototype of enumeration of nonsolvable groups, extending the work of [BEO02] to the nonsolvable case. It illustrates the feasibility range of current implementations of underlying routines for cohomology, extensions, and isomorphism tests, with a number of general-purpose improvements in the system GAP [GAP20] (that will be part of the 4.12 release) by the author having been motivated by this work. Indeed, the fact, that it took over 30 years since the publication of [HP89] to complete the classification of perfect groups up to order one million, indicates the broad infrastructural 1 ar X iv :2 10 4. 10 82 8v 3 [ m at h. G R ] 6 J ul 2 02 1 requirements of such classifications, wi","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77334474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2007, Grytczuk conjecture that for any sequence (li)i≥1 of alphabets of size 3 there exists a square-free infinite word w such that for all i, the ith letter of w belongs to li. The result of Thue of 1906 implies that there is an infinite square-free word if all the li are identical. On the other, hand Grytczuk, Przyby lo and Zhu showed in 2011 that it also holds if the li are of size 4 instead of 3. In this article, we first show that if the lists are of size 4, the number of square-free words is at least 2.45 (the previous similar bound was 2). We then show our main result: we can construct such a square-free word if the lists are subsets of size 3 of the same alphabet of size 4. Our proof also implies that there are at least 1.25 square-free words of length n for any such list assignment. This proof relies on the existence of a set of coefficients verified with a computer. We suspect that the full conjecture could be resolved by this method with a much more powerful computer (but we might need to wait a few decades for such a computer to be available).
{"title":"Avoiding squares over words with lists of size three amongst four symbols","authors":"M. Rosenfeld","doi":"10.1090/mcom/3732","DOIUrl":"https://doi.org/10.1090/mcom/3732","url":null,"abstract":"In 2007, Grytczuk conjecture that for any sequence (li)i≥1 of alphabets of size 3 there exists a square-free infinite word w such that for all i, the ith letter of w belongs to li. The result of Thue of 1906 implies that there is an infinite square-free word if all the li are identical. On the other, hand Grytczuk, Przyby lo and Zhu showed in 2011 that it also holds if the li are of size 4 instead of 3. In this article, we first show that if the lists are of size 4, the number of square-free words is at least 2.45 (the previous similar bound was 2). We then show our main result: we can construct such a square-free word if the lists are subsets of size 3 of the same alphabet of size 4. Our proof also implies that there are at least 1.25 square-free words of length n for any such list assignment. This proof relies on the existence of a set of coefficients verified with a computer. We suspect that the full conjecture could be resolved by this method with a much more powerful computer (but we might need to wait a few decades for such a computer to be available).","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77647708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory. This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.
{"title":"On the construction of multiresolution analyses associated to general subdivision schemes","authors":"Zhiqing Kui, J. Baccou, J. Liandrat","doi":"10.1090/MCOM/3646","DOIUrl":"https://doi.org/10.1090/MCOM/3646","url":null,"abstract":"Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory.\u0000\u0000This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86090528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Continued fractions have been widely studied in the field of pp-adic numbers Qpmathbb Q_p, but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call standard, for any quadratic irrational via pp-adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in Rmathbb R and Qpmathbb Q_p. Moreover given two primes p1p_1 and p2
连分式在p进数Q p mathbb Q_p领域得到了广泛的研究，但目前还没有一种算法能复制连分式在实数上所具有的所有优良性质，特别是在有限性和周期性方面。在本文中，我们首先提出了一个周期表示，我们称之为标准，对于任何二次无理数通过p进连分数，即使它不是由一个特定的算法得到。这个周期表示提供了R mathbb R和Q p mathbb Q_p中二次无理数的同时有理逼近。此外，给定两个素数p1p_1和p2p_2，利用二项式变换，我们也可以将qp1 mathbb {Q}_{p_1}中的近似传递到给定二次无理数的qp2 mathbb {Q}_{p_2}中的近似。然后，我们将重点放在一个特定的p进连分数算法上，证明它在处理有理数时停止在有限的步骤中，解决了Browkin [Math]的一篇论文中留下的一个问题。[p. 70(2001)，第1281-1292页]。最后，我们研究了该算法的周期性，表明它何时产生二次无理数的标准表示。
{"title":"Periodic representations for quadratic irrationals in the field of p-adic numbers","authors":"Stefano Barbero, Umberto Cerruti, N. Murru","doi":"10.1090/MCOM/3640","DOIUrl":"https://doi.org/10.1090/MCOM/3640","url":null,"abstract":"<p>Continued fractions have been widely studied in the field of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic numbers <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q Subscript p\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb Q_p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call <italic>standard</italic>, for any quadratic irrational via <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q Subscript p\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb Q_p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Moreover given two primes <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">p_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p 2\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 ","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86501769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Isotropic non-Lipschitz regularization for sparse representations of random fields on the sphere","authors":"Chao Li, Xiaojun Chen","doi":"10.1090/MCOM/3655","DOIUrl":"https://doi.org/10.1090/MCOM/3655","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85615812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. A. Martínez, S. Imperiale, P. Joly, Jerónimo Rodríguez
When solving 2D linear elastodynamic equations in a homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated In [3, 2]. However in [4] the case of free surface boundary condition was proven to be unstable if a straightforward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented in [4] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown.
{"title":"Numerical analysis of a method for solving 2D linear isotropic elastodynamics with traction free boundary condition using potentials and finite elements","authors":"J. A. Martínez, S. Imperiale, P. Joly, Jerónimo Rodríguez","doi":"10.1090/mcom/3613","DOIUrl":"https://doi.org/10.1090/mcom/3613","url":null,"abstract":"When solving 2D linear elastodynamic equations in a homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated In [3, 2]. However in [4] the case of free surface boundary condition was proven to be unstable if a straightforward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented in [4] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75916353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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