a priori error analysis - lebesgue...second-order pdes of mixed type 157 considerthepdeproblem:find...

10
A Priori Error Analysis 149 Key idea: select I p = Π L 2 and assume the following holds. Assumption 6 Assume that the advection field b satisfies the following condition: b · h ξ V p (T h ) ξ V p (T h ). Hence, B ar (η , ξ ) = κT h 2 κ (c 2 0 - c)ξη dx - - κ\Ω b · n κ (ξ + - ξ - )η - ds - + κΩ b · n κ η + ξ + ds , since κ η b · ξ dx κ (u - Π L 2 u)b · ξ dx = 0 ξ V p (T h ).

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  • A Priori Error Analysis

    149

    � Key idea: select Ip = �L2 and assume the following holds.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

    Assumption 6

    Assume that the advection field b satisfies the following condition:

    b · �h� � Vp(Th) �� � Vp(Th).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

    Hence,

    Bar(�, �) =�

    ��Th

    �2

    �(c20 � c)��dx

    ��

    ���\��b · n� (�+ � ��)�� ds �

    �+����b · n� �+�+ ds

    �,

    since �

    ��b · ��dx �

    �(u � �L2u)b · ��dx = 0 �� � Vp(Th).

    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

  • A Priori Error Analysis

    150

    On standard element shapes, optimal L2-approximation bounds are availablefor both the volume and face terms; hence

    |�u � uh|�DG � Chs�1/2

    pl�1/2�u�Hl(�).

    AAAEA3icbVLLbtNAFHUaHiW8UliyYESD1EpJiMMCJCRUqUjtolKLoA+pTqLr8XU8ynhmmBlXqVwv+Rp2iC1/wA/wN4ydINVp7+rMue8zN1ScGTsY/G2sNe/cvXd//UHr4aPHT562N56dGJlpisdUcqnPQjDImcBjyyzHM6UR0pDjaTjbLf2nF6gNk+KrvVQ4SmEqWMwoWEdN2n8CIZmIUFhyKIixICLQEUGOacmZBBSaLpHKsh 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AAAEA3icbVLLbtNAFHUaHiW8UliyYESD1EpJiMMCJCRUqUjtolKLoA+pTqLr8XU8ynhmmBlXqVwv+Rp2iC1/wA/wN4ydINVp7+rMue8zN1ScGTsY/G2sNe/cvXd//UHr4aPHT562N56dGJlpisdUcqnPQjDImcBjyyzHM6UR0pDjaTjbLf2nF6gNk+KrvVQ4SmEqWMwoWEdN2n8CIZmIUFhyKIixICLQEUGOacmZBBSaLpHKsh 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AAAEA3icbVLLbtNAFHUaHiW8UliyYESD1EpJiMMCJCRUqUjtolKLoA+pTqLr8XU8ynhmmBlXqVwv+Rp2iC1/wA/wN4ydINVp7+rMue8zN1ScGTsY/G2sNe/cvXd//UHr4aPHT562N56dGJlpisdUcqnPQjDImcBjyyzHM6UR0pDjaTjbLf2nF6gNk+KrvVQ4SmEqWMwoWEdN2n8CIZmIUFhyKIixICLQEUGOacmZBBSaLpHKsh 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AAAEA3icbVLLbtNAFHUaHiW8UliyYESD1EpJiMMCJCRUqUjtolKLoA+pTqLr8XU8ynhmmBlXqVwv+Rp2iC1/wA/wN4ydINVp7+rMue8zN1ScGTsY/G2sNe/cvXd//UHr4aPHT562N56dGJlpisdUcqnPQjDImcBjyyzHM6UR0pDjaTjbLf2nF6gNk+KrvVQ4SmEqWMwoWEdN2n8CIZmIUFhyKIixICLQEUGOacmZBBSaLpHKsh 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

    H., Schwab & Suli 2002, Chernov 2012, Melenk & Wurzer 2014

    • Alternative: Introduce the projection of bbb onto the space of piecewiseconstant functions

    � Leads to suboptimal hp-bound.AAAD7nicbVJbaxNBFN40Xmq8NNVHEQcboUIakvqg+CCVClWoUC9pC91QZmfPJmNmZ4aZs23Ksn/DN/HVf+Bv8d94dhOhm/bAwrff+c51TmSV9Njv/22sNG/cvHV79U7r7r37D9ba6w8PvcmcgKEwyrjjiHtQUsMQJSo4tg54Gik4iqa7pf/oDJyXRn/DCwujlI+1TKTgSNRp+0+ojdQxaGR5GEesE0aZUoCdgr1TCE6T7gzesI 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AAAD7nicbVJbaxNBFN40Xmq8NNVHEQcboUIakvqg+CCVClWoUC9pC91QZmfPJmNmZ4aZs23Ksn/DN/HVf+Bv8d94dhOhm/bAwrff+c51TmSV9Njv/22sNG/cvHV79U7r7r37D9ba6w8PvcmcgKEwyrjjiHtQUsMQJSo4tg54Gik4iqa7pf/oDJyXRn/DCwujlI+1TKTgSNRp+0+ojdQxaGR5GEesE0aZUoCdgr1TCE6T7gzesI 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AAAD7nicbVJbaxNBFN40Xmq8NNVHEQcboUIakvqg+CCVClWoUC9pC91QZmfPJmNmZ4aZs23Ksn/DN/HVf+Bv8d94dhOhm/bAwrff+c51TmSV9Njv/22sNG/cvHV79U7r7r37D9ba6w8PvcmcgKEwyrjjiHtQUsMQJSo4tg54Gik4iqa7pf/oDJyXRn/DCwujlI+1TKTgSNRp+0+ojdQxaGR5GEesE0aZUoCdgr1TCE6T7gzesI 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AAAD7nicbVJbaxNBFN40Xmq8NNVHEQcboUIakvqg+CCVClWoUC9pC91QZmfPJmNmZ4aZs23Ksn/DN/HVf+Bv8d94dhOhm/bAwrff+c51TmSV9Njv/22sNG/cvHV79U7r7r37D9ba6w8PvcmcgKEwyrjjiHtQUsMQJSo4tg54Gik4iqa7pf/oDJyXRn/DCwujlI+1TKTgSNRp+0+ojdQxaGR5GEesE0aZUoCdgr1TCE6T7gzesI 8anYkzAQwnwKwz30GUGZhJKOYMRB4VHWY0mkrgLScp+awEAefSAwtDJoz2yKlUkukq2hPbauWhg5iyfJHjCXLnzDlV3wcee0bpfBYZizLlinUmtrMVmUzHvdP2Rr/Xr4xdBYMF2AgWdnC6vvI0jI3IUhpVKO79yaBvcZRzh1IoKFph5oG6nvIxnBDUPAU/yqsNF+w5MTFLjKOP+q/YyxE5T72/SCNSphwnftlXktf5TjJMXo9yqW2GoMW8UJKpcvDyuVgsHS1aXRDgwknqlYkJd1zQ29SraCkgIUdtknzsuJ1IMauzHIprO6yRgitxRVYNWSOtRyfF1HcJbGkTQ5eVCGGGc/QyrgdUS7CwlHuW0QAUvcQqnKHjRHrAlEtdrj/fk0qxr1z7/zwlLB2b7+VYou/u09Hq7p4DmL6oiS9ljlL6r/ZODQlnfEEnNVg+oKvgcLs36PcGn7c3dt4ujms1eBw8CzaDQfAq2Ak+BAfBMBCNJ43dxn7jU9M2fzR/Nn/NpSuNRcyjoGbN3/8AZCRQHQ==

    • Optimal L2 bounds on polytopic elements not available.AAADinicbVJdaxNBFJ1m/ai1aqqPPjiYCBVi2I0PKoIULNSHgpWatpDEcHf2JhkyO7PM3C0JS36YP8UnX/VfOJtE6Ca9MHDm3HvuFzfOlHQUhr92asGdu/fu7z7Ye7j/6PGT+sHTC2dyK7ArjDL2KgaHSmrskiSFV5lFSGOFl/H0c+m/vEbrpNHfaZ7hIIWxliMpgDw1rJ/3tZE6QU286Ccxb/bjXCmk5oJ/zUimoHjz9EenyW 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 OT68Rxo3lm1JxMJgVHhalXOq4NcbgGqcDXbQ/rjbAdLo1vg2gNGmxtZ8OD2ot+YkReJhMKnOtFYUaDAixJoXCx188dZiCmMMaehxpSdINiOf2Cv/JMwkfG+ufHWLI3FQWkzs3T2EemQBO36SvJ23y9nEbvB4XUWU6oxarQKFecDC9XyRNpUZCaewDCSt8rFxOwIMgvvFJFS4Ej76hMUowtZBMpZlUWcHFrhxVSgBJbYcshK2TmyEoxdS0P3miTYIuXiHBGK/Q2qQqWS8hwI/cs9wN49QaraEYWPOmQUpC6XH9xIpXi56Ddf94nLB2Hx3IsybVO/eHp1olFnL6uBN/IHKf+v9y7b0hY4xb+pKLNA9oGF512FLajb53G0af1ce2y5+wlO2QRe8eO2Bd2xrpMsJ/sN/vD/gb7QSf4EHxchdZ21ppnrGLB8T/vdiwcAAADinicbVJdaxNBFJ1m/ai1aqqPPjiYCBVi2I0PKoIULNSHgpWatpDEcHf2JhkyO7PM3C0JS36YP8UnX/VfOJtE6Ca9MHDm3HvuFzfOlHQUhr92asGdu/fu7z7Ye7j/6PGT+sHTC2dyK7ArjDL2KgaHSmrskiSFV5lFSGOFl/H0c+m/vEbrpNHfaZ7hIIWxliMpgDw1rJ/3tZE6QU286Ccxb/bjXCmk5oJ/zUimoHjz9EenyW OT68Rxo3lm1JxMJgVHhalXOq4NcbgGqcDXbQ/rjbAdLo1vg2gNGmxtZ8OD2ot+YkReJhMKnOtFYUaDAixJoXCx188dZiCmMMaehxpSdINiOf2Cv/JMwkfG+ufHWLI3FQWkzs3T2EemQBO36SvJ23y9nEbvB4XUWU6oxarQKFecDC9XyRNpUZCaewDCSt8rFxOwIMgvvFJFS4Ej76hMUowtZBMpZlUWcHFrhxVSgBJbYcshK2TmyEoxdS0P3miTYIuXiHBGK/Q2qQqWS8hwI/cs9wN49QaraEYWPOmQUpC6XH9xIpXi56Ddf94nLB2Hx3IsybVO/eHp1olFnL6uBN/IHKf+v9y7b0hY4xb+pKLNA9oGF512FLajb53G0af1ce2y5+wlO2QRe8eO2Bd2xrpMsJ/sN/vD/gb7QSf4EHxchdZ21ppnrGLB8T/vdiwc

  • A Priori Error Analysis: Polytopic Meshes

    151

    An alternative approach is based on establishing an inf-sup condition for thebilinear form Bar(·, ·), with respect to the following streamline-diffusion DGFEMnorm:

    |�v|�s :=�|�v|�2DG +

    ��Th

    ���b · �v�2L2(�)�1/2

    ,

    where

    �� :=1

    �b�L�(�)h��p2�

    �� � Th,

    for p� � 1.AAAFXnicbVPdjtQ2FA5TBmhaCpSbSr3oUXepZsXsMpleFK1ERYF2e0ElqnYBab2zcpyTxBrHDrazP4ryCDwNt+2DcNdH6XFmtiW7WFF0/Pn8fOf4c1or6fxs9v7K6JOr42vXb3waf/b5zS9u3b7z5UtnGitwXxhl7OuUO1RS476XXuHr2iKvUoWv0uXTcP7qGK2TRv/pz2o8rHihZS4F9wQd3Rl9x7SROkPtAX7SwJVHq+nwGI 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

    The mesh parameter h�� is defined as follows:

    h�� := minF���

    sup�F��� |�F�||F| d �� � T, d = 2, 3,

    Note that the following relation holds: h�� � h�.AAAEjnicbVJtb9MwEE5HgRHeNvjIB06sSEMqVbtJgCYNJoHYPiA0xN6kZasc59JYdezMdqBT1v0zfgif+Qr/gUvawdLNUuzLc8+d73xPmElhXbf7szF3o3nz1u35O/7de/cfPFxYfLRndW447nIttTkImUUpFO464SQeZAZZGkrcD4fvS//+NzRWaLXjTjM8StlAiVhw5gjqLzb2AqWFilA5gJ0EIUWb+BkzLEWHBlpJPxiyLG 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AAAEjnicbVJtb9MwEE5HgRHeNvjIB06sSEMqVbtJgCYNJoHYPiA0xN6kZasc59JYdezMdqBT1v0zfgif+Qr/gUvawdLNUuzLc8+d73xPmElhXbf7szF3o3nz1u35O/7de/cfPFxYfLRndW447nIttTkImUUpFO464SQeZAZZGkrcD4fvS//+NzRWaLXjTjM8StlAiVhw5gjqLzb2AqWFilA5gJ0EIUWb+BkzLEWHBlpJPxiyLG 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AAAEjnicbVJtb9MwEE5HgRHeNvjIB06sSEMqVbtJgCYNJoHYPiA0xN6kZasc59JYdezMdqBT1v0zfgif+Qr/gUvawdLNUuzLc8+d73xPmElhXbf7szF3o3nz1u35O/7de/cfPFxYfLRndW447nIttTkImUUpFO464SQeZAZZGkrcD4fvS//+NzRWaLXjTjM8StlAiVhw5gjqLzb2AqWFilA5gJ0EIUWb+BkzLEWHBlpJPxiyLG 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AAAEjnicbVJtb9MwEE5HgRHeNvjIB06sSEMqVbtJgCYNJoHYPiA0xN6kZasc59JYdezMdqBT1v0zfgif+Qr/gUvawdLNUuzLc8+d73xPmElhXbf7szF3o3nz1u35O/7de/cfPFxYfLRndW447nIttTkImUUpFO464SQeZAZZGkrcD4fvS//+NzRWaLXjTjM8StlAiVhw5gjqLzb2AqWFilA5gJ0EIUWb+BkzLEWHBlpJPxiyLG 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

    Ayuso & Marini 2009, Buffa, et al. 2006, Cangiani et al. 2013, Johnson & Pitkaranta 1986

  • A Priori Error Analysis: Polytopic Meshes

    152

    Theorem 3

    • The mesh is assumed to be shape regular, since the proof relies onemploying an inverse inequality of the form

    ��v�2L2(�) � Cinv,5p4

    h2��v�2L2(�) �v � Pp(�),

    where Cinv,5 > 0 is independent of v, h�, and p.

    • Assumption 2 requires the number of element faces to be uniformlybounded.

    AAAFGnicbVTLbhMxFJ2GACW8WliywKJBaqUhSgIIVlVRkQpSF0X0JXXSyOO5yVjx2K7tCQnT+RO+hh1iy4YtX8J1HlKT1lKUq/s899yjibXg1jWbf1cqt6q379xdvVe7/+Dho8dr60+OrcoNgyOmhDKnMbUguIQjx52AU22AZrGAk3iw6+MnQzCWK3noxho6Ge1L3uOMOnR111f+RTH0uSy4g4x/h7IWeYscpkAysCnhllBr8wwS4hSJgdiUaiAG+rmgJiSWSwbEYbo2SvUwIDhYoiSJIgKZFmrMZZ9QSbj0QAD/4SKngrsxwXxf2VMmq81w+JiHhkAuI0ljQckwuuwW++ftzWhAtaZb5XmbkEgAIbvdoohM5luX4duSRD1DWaHP35RF2i2m6ZhdRhInFMNyqUstusBpCZYpQ4XAQVxGGXUpo6I4KLt6nhvWIilVLQKZXAH4LQUDpD4HMUex3ax71rhMQGMBSOf3rA/rYa2edqcd6yEykpC6rjdmhH/wJGvfmLSRxIucG6TRsyPzLAbje4CAzLfrUeZjk3PkeE1cTow937HKcWDSmCKdn7S7ttFsNCePXDdaM2MjmL2D7nrleZQolvthTOD5z1pN7ToFNY4z4TWSW9CUDWgfztCUFKXSKSZqLMlL9CT+pvhDsBPv1YqCZtaOsxgzPdd2OeadN8XOctd73ym41LkDyaaDernwRHhpkwQpYw6ZSDhlhiNWwlKKinCou4UpkjPwUlnYpOgbqlPORoteCuWNCBecKBh2LW2y5IJTW2c4G9gQjVdSJRASbzkYuan1OlksmJCgYan3CI/OsHrJK9zIGYpOCy6jXHr6iz2Owv5KpZ37saEPbH7kfe5suI9iluGeARhsLSRf6Rxn/rPgeUdAzCjrJdVaFtB147jdaDUbrS/tjZ3tmbhWg2fBi2AzaAXvgp3gU3AQHAWs8rmiKqPKuPqj+rP6q/p7mlpZmdU8DRZe9c9/gdi9og==AAAFGnicbVTLbhMxFJ2GACW8WliywKJBaqUhSgIIVlVRkQpSF0X0JXXSyOO5yVjx2K7tCQnT+RO+hh1iy4YtX8J1HlKT1lKUq/s899yjibXg1jWbf1cqt6q379xdvVe7/+Dho8dr60+OrcoNgyOmhDKnMbUguIQjx52AU22AZrGAk3iw6+MnQzCWK3noxho6Ge1L3uOMOnR111f+RTH0uSy4g4x/h7IWeYscpkAysCnhllBr8wwS4hSJgdiUaiAG+rmgJiSWSwbEYbo2SvUwIDhYoiSJIgKZFmrMZZ9QSbj0QAD/4SKngrsxwXxf2VMmq81w+JiHhkAuI0ljQckwuuwW++ftzWhAtaZb5XmbkEgAIbvdoohM5luX4duSRD1DWaHP35RF2i2m6ZhdRhInFMNyqUstusBpCZYpQ4XAQVxGGXUpo6I4KLt6nhvWIilVLQKZXAH4LQUDpD4HMUex3ax71rhMQGMBSOf3rA/rYa2edqcd6yEykpC6rjdmhH/wJGvfmLSRxIucG6TRsyPzLAbje4CAzLfrUeZjk3PkeE1cTow937HKcWDSmCKdn7S7ttFsNCePXDdaM2MjmL2D7nrleZQolvthTOD5z1pN7ToFNY4z4TWSW9CUDWgfztCUFKXSKSZqLMlL9CT+pvhDsBPv1YqCZtaOsxgzPdd2OeadN8XOctd73ym41LkDyaaDernwRHhpkwQpYw6ZSDhlhiNWwlKKinCou4UpkjPwUlnYpOgbqlPORoteCuWNCBecKBh2LW2y5IJTW2c4G9gQjVdSJRASbzkYuan1OlksmJCgYan3CI/OsHrJK9zIGYpOCy6jXHr6iz2Owv5KpZ37saEPbH7kfe5suI9iluGeARhsLSRf6Rxn/rPgeUdAzCjrJdVaFtB147jdaDUbrS/tjZ3tmbhWg2fBi2AzaAXvgp3gU3AQHAWs8rmiKqPKuPqj+rP6q/p7mlpZmdU8DRZe9c9/gdi9og==AAAFGnicbVTLbhMxFJ2GACW8WliywKJBaqUhSgIIVlVRkQpSF0X0JXXSyOO5yVjx2K7tCQnT+RO+hh1iy4YtX8J1HlKT1lKUq/s899yjibXg1jWbf1cqt6q379xdvVe7/+Dho8dr60+OrcoNgyOmhDKnMbUguIQjx52AU22AZrGAk3iw6+MnQzCWK3noxho6Ge1L3uOMOnR111f+RTH0uSy4g4x/h7IWeYscpkAysCnhllBr8wwS4hSJgdiUaiAG+rmgJiSWSwbEYbo2SvUwIDhYoiSJIgKZFmrMZZ9QSbj0QAD/4SKngrsxwXxf2VMmq81w+JiHhkAuI0ljQckwuuwW++ftzWhAtaZb5XmbkEgAIbvdoohM5luX4duSRD1DWaHP35RF2i2m6ZhdRhInFMNyqUstusBpCZYpQ4XAQVxGGXUpo6I4KLt6nhvWIilVLQKZXAH4LQUDpD4HMUex3ax71rhMQGMBSOf3rA/rYa2edqcd6yEykpC6rjdmhH/wJGvfmLSRxIucG6TRsyPzLAbje4CAzLfrUeZjk3PkeE1cTow937HKcWDSmCKdn7S7ttFsNCePXDdaM2MjmL2D7nrleZQolvthTOD5z1pN7ToFNY4z4TWSW9CUDWgfztCUFKXSKSZqLMlL9CT+pvhDsBPv1YqCZtaOsxgzPdd2OeadN8XOctd73ym41LkDyaaDernwRHhpkwQpYw6ZSDhlhiNWwlKKinCou4UpkjPwUlnYpOgbqlPORoteCuWNCBecKBh2LW2y5IJTW2c4G9gQjVdSJRASbzkYuan1OlksmJCgYan3CI/OsHrJK9zIGYpOCy6jXHr6iz2Owv5KpZ37saEPbH7kfe5suI9iluGeARhsLSRf6Rxn/rPgeUdAzCjrJdVaFtB147jdaDUbrS/tjZ3tmbhWg2fBi2AzaAXvgp3gU3AQHAWs8rmiKqPKuPqj+rP6q/p7mlpZmdU8DRZe9c9/gdi9og==AAAFGnicbVTLbhMxFJ2GACW8WliywKJBaqUhSgIIVlVRkQpSF0X0JXXSyOO5yVjx2K7tCQnT+RO+hh1iy4YtX8J1HlKT1lKUq/s899yjibXg1jWbf1cqt6q379xdvVe7/+Dho8dr60+OrcoNgyOmhDKnMbUguIQjx52AU22AZrGAk3iw6+MnQzCWK3noxho6Ge1L3uOMOnR111f+RTH0uSy4g4x/h7IWeYscpkAysCnhllBr8wwS4hSJgdiUaiAG+rmgJiSWSwbEYbo2SvUwIDhYoiSJIgKZFmrMZZ9QSbj0QAD/4SKngrsxwXxf2VMmq81w+JiHhkAuI0ljQckwuuwW++ftzWhAtaZb5XmbkEgAIbvdoohM5luX4duSRD1DWaHP35RF2i2m6ZhdRhInFMNyqUstusBpCZYpQ4XAQVxGGXUpo6I4KLt6nhvWIilVLQKZXAH4LQUDpD4HMUex3ax71rhMQGMBSOf3rA/rYa2edqcd6yEykpC6rjdmhH/wJGvfmLSRxIucG6TRsyPzLAbje4CAzLfrUeZjk3PkeE1cTow937HKcWDSmCKdn7S7ttFsNCePXDdaM2MjmL2D7nrleZQolvthTOD5z1pN7ToFNY4z4TWSW9CUDWgfztCUFKXSKSZqLMlL9CT+pvhDsBPv1YqCZtaOsxgzPdd2OeadN8XOctd73ym41LkDyaaDernwRHhpkwQpYw6ZSDhlhiNWwlKKinCou4UpkjPwUlnYpOgbqlPORoteCuWNCBecKBh2LW2y5IJTW2c4G9gQjVdSJRASbzkYuan1OlksmJCgYan3CI/OsHrJK9zIGYpOCy6jXHr6iz2Owv5KpZ37saEPbH7kfe5suI9iluGeARhsLSRf6Rxn/rPgeUdAzCjrJdVaFtB147jdaDUbrS/tjZ3tmbhWg2fBi2AzaAXvgp3gU3AQHAWs8rmiKqPKuPqj+rP6q/p7mlpZmdU8DRZe9c9/gdi9og==

    Let Th = {�} be a subdivision of � � Rd, d = 2, 3, consisting of shape regularpolytopic elements satisfying Assumption 2. Then, assuming that b satisfiesAssumption 6, there exists a positive constant �s, independent of the meshsize h and the polynomial degree p, such that:

    inf��Vp(Th)\{0}

    supµ�Vp(Th)\{0}

    Bar(�, µ)|��|�s|�µ|�s

    � �s.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

  • A Priori Error Analysis: Polytopic Meshes

    153

    Lemma 15

    Proof:

    = 0 (by G.O.)

    |�u � uh|�s � |�u � �̃pppu|�s + |��̃pppu � uh|�s.

    Exploiting the inf-sup condition gives

    |��̃pppu � uh|�s �1�s

    sup�h�Vp(Th)\{0}

    Bar(�̃pppu � uh, �h)|��h|�s

    � 1�s

    sup�h�Vp(Th)\{0}

    |Bar(�̃pppu � u, �h)||��h|�s

    +1�s

    sup�h�Vp(Th)\{0}

    |Bar(u � uh, �h)||��h|�s

    .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

    Under the hypotheses of Theorem 3, the following bound holds

    |�u � uh|�s � |�u � �̃pppu|�s +1�s

    sup�h�Vp(Th)\{0}

    |Bar(�̃pppu � u, �h)||��h|�s

    where �̃ppp|� := �̃p� for all � � 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

  • A Priori Error Analysis: Polytopic Meshes

    154

    Theorem 4Let Th = {�} be a subdivision of � � Rd, d = 2, 3, satisfying the hypotheses ofTheorem 3. Assuming u � H1(�) satisfies u|� � Hl�(�), l� > 1 + d/2, for each� � Th, such that Eu|K � Hl�(K), where K � T �h with � � K, then

    |�u � uh|�2s � C�

    ��Th

    h2s��p2l��

    �G�(F, Cm, p�, ��)�Eu�2Hl� (K)

    �,

    where

    G�(F, Cm, p�, ��) := �c0�2L�(�) + �2� + ��1� + ���2�p2�h�2�+ ��p�h�d�

    F���Cm(p�, �, F)|F|,

    and s� = min{p� + 1, l�}, �� := �c/c0�L�(�), �� := �b�L�(�), and C is a positiveconstant, which depends on the shape-regularity of T �h , but is independent of thediscretization parameters.

    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

  • A Priori Error Analysis: Polytopic Meshes

    155

    For uniform orders p� = p � 1, h = max��Th h�, s� = s, s = min{p + 1, l},l > 1 + d/2, under the assumption that the diameter of the faces of each element� � Th is of comparable size to the diameter of the corresponding element, i.e.,diam(F) � h�, h�� � h�, F � ��, � � Th, so that |F| � h

    (d�1)� the a priori error bound

    yields

    |�u � uh|�DG � |�u � uh|�s � Chs�

    12

    pl�1�u�Hl(�).

    Hence, the above hp-bound is optimal in h and suboptimal in p by p1/2.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

  • Outline

    DGFEMs on Polytopic Meshes II: Hyperbolic PDEsExtension to Second-Order PDEs of Mixed Type

    156

  • Second-Order PDEs of Mixed Type

    157

    Consider the PDE problem: find u such that

    �� · (a�u) + � · (bbbu) + cu = f in �,u = gD on ��D � ���,

    n · (a�u) = gN on ��N;

    here, f � L2(�), a =�aij

    �di,j=1, with aij � L

    �(�), aij = aji, for i, j = 1, . . . , d,

    such that, at each x in �̄, we have

    d�

    i,j=1

    aij(x)�i�j � 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

    Here, �0� = ��D � ��N, where

    �0� :=�x � �� :

    d�

    i,j=1

    aij(x)ninj > 0�

    ,

    ��� := {x � ��\�0� : bbb(x) · nnn(x) < 0} ,�+� := {x � ��\�0� : bbb(x) · nnn(x) � 0} .

    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

    H. & Suli 2000, H. Schwab & Suli 2002

  • DGFEM Approximation

    158

    Find uh � Vp(Th) such that�

    �a�huh · �hvh + �h · (bbbuh)vh + cuh vh

    �dx

    +

    FIh �FhD

    �� {{a�huh}} · [[vh]] � {{a�hvh}} · [[uh]] + �[[uh]] · [[vh]]

    �ds

    ��

    ��Th

    ���\��b · n� (u+h � u

    �h )v

    +h ds �

    ��Th

    ����(��D����)b · n u+h v

    +h ds

    =

    �f vhdx +

    ��D

    gD(�a�hvh · n + �vh)ds +�

    ��N

    gNvhds

    ��

    ��Th

    ����(��D����)b · n gDv+h ds

    for all vh � Vp(Th).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