Projection in negative norms and the regularization of rough linear functionals
Projection in negative norms and the regularization of rough linear functionals
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DOI：
10.1007/s00211-022-01278-z
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发表时间：
2021-01
影响因子：
2.1
通讯作者：
Felipe Millar;I. Muga;Sergio Rojas;K. Zee
中科院分区：
文献类型：
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作者：
Felipe Millar;I. Muga;Sergio Rojas;K. Zee
为了构造作用在Sobolev空间上的连续线性泛函的正则化，其中和是Lipschitz域，我们在负Sobolev空间中提出了一种投影方法，p是共轭指数，满足.我们的方法是特别有用的，当一个是处理一个粗糙（不规则）的功能，是一个成员，虽然不是，但一个努力的正规近似。我们专注于投影到离散有限元空间，并考虑不连续以及连续分段多项式逼近。虽然所提出的方法旨在计算在负（对偶）范数中测量的最佳近似，但出于实际原因，我们将采用在离散子空间上最大化的可计算的离散对偶范数。我们表明，这种想法导致一个完全离散的方法给出的混合问题。我们提出了一个不连续的，以及一个连续的最低阶对，证明他们是兼容的，因此获得准最优收敛方法。我们目前的数值实验，计算狄拉克三角洲和线源的有限元近似。我们还提出了自适应生成网格，从误差表示，与该方法。最后，我们展示了如何提出的投影方法可以用来有效地计算数值近似的偏微分方程与粗糙的数据。
In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as, whereandis a Lipschitz domain, we propose a projection method in negative Sobolev spaces,pbeing the conjugate exponent satisfying. Our method is particularly useful when one is dealing with a rough (irregular) functional that is a member of, though not of, but one strives for a regular approximation in. We focus on projections onto discrete finite element spaces, and consider both discontinuous as well as continuous piecewise-polynomial approximations. While the proposed method aims to compute the best approximation as measured in the negative (dual) norm, for practical reasons, we will employ a computable, discrete dual norm that supremizes over a discrete subspace. We show that this idea leads to a fully discrete method given by a mixed problem on. We propose a discontinuous as well as a continuous lowest-order pair, prove that they are compatible, and therefore obtain quasi-optimally convergent methods . We present numerical experiments that compute finite element approximations to Dirac delta’s and line sources. We also present adaptively generate meshes, obtained from an error representation that comes with the method. Finally, we show how the presented projection method can be used to efficiently compute numerical approximations to partial differential equations with rough data.