Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations

Efficient Preconditioners for Saddle Point Systems

Author(s): '> Zhi-Hao Cao

Pp: 23-43 (21)

DOI: 10.2174/978160805291211101010023

* (Excluding Mailing and Handling)

Abstract

Various block preconditioners for two by two block linear saddle point systems are studied. All block preconditioners are derived from a splitting of the (1,1) block of the two by two block matrix. We analyze the properties of the corresponding preconditioned matrices, in particular their spectra, and discuss the computational performance of the preconditioned iterative methods. It is shown that fast convergence depends mainly on the quality of the splitting of the (1,1) block. Moreover, some strategies of the implementation of the block preconditioners based on purely algebraic considerations are discussed. Thus, applying our block preconditioners to the related saddle point problems, we obtain preconditioned iterative methods in a “black box” fashion.


Keywords: saddle point systems, iterative methods, preconditioning, convergence, algebraic, eigenvalues, block preconditioners

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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1518
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 860 and `b`.`banner_from_date` <= 1739834601 and `b`.`banner_to_date` >= 1739834601 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.06msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 860
      • 3: 1739834601
      • 4: 1739834601
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='bd56c085dafe9b4ae0726412601516fc'
      200μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    7Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        e8pyG5P6LGCEJMiadc63335KljUP82PdFbbB474d
        uc_cart_id
        bd56c085dafe9b4ae0726412601516fc
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/930" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/930
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "wayfindertarget=eyJpdiI6ImdKb2dYL2F1QThIQ00zUmpOb1diaXc9PSIsInZhbHVlIjoiUWl3QVZZVzBNTjdjVE54UEtoWElUaTE2bHpLSGd1NFRuMnZiNkNXTnpaYz0iLCJtYWMiOiI2NWQyYWJiM2U0OWM0YTNlYTJmMWYyZmJmYTRlYjIxMjY3MTJkZDFkNTc0OTBkYWI1M2I0OGJlZjI5ZmNkMTE4IiwidGFnIjoiIn0%3Dwayfindertarget=eyJpdiI6ImdKb2dYL2F1QThIQ00zUmpOb1diaXc9PSIsInZhbHVlIjoiUWl3QVZZVzBNTjdjVE54UEtoWElUaTE2bHpLSGd1NFRuMnZiNkNXTnpaYz0iLCJtYWMiOiI2NWQyYWJiM2U0OWM0" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
        array:1 [ "wayfindertarget" => null ]
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Mon, 17 Feb 2025 23:23:21 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InNHSk0veXVPUEw0c2tScUN4V2szN0E9PSIsInZhbHVlIjoiSkJkQmZrQmdLY3dFZGZzYk1BanpvK0RERGtMSEFFSmY5Zlk1OE9sTHE5NFZRREp2VER4YlRsLzMxQUNNd3VNRFBVbE4xM3VPamVXcmJ2NDZGYndlRDczZVlsenA4bDR0dEtFekUrcHFlQS92Unh4SldFZi8xM25qOFhZLzNkMUsiLCJtYWMiOiI5N2FlYThlODZlNTVhNGMyYmEwZGE0MmQ5MzliZGIxMDk4N2VkZDg2ZmQzZmE5MGIxMGExZTU2YTg4MjFmOGY1IiwidGFnIjoiIn0%3D; expires=Thu, 20 Feb 2025 11:23:21 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6InNHSk0veXVPUEw0c2tScUN4V2szN0E9PSIsInZhbHVlIjoiSkJkQmZrQmdLY3dFZGZzYk1BanpvK0RERGtMSEFFSmY5Zlk1OE9sTHE5NFZRREp2VER4YlRsLzMxQUNNd3VNRFBVbE4xM" 1 => "alphaeurekaselectnet_session=eyJpdiI6IklLb3dvN2NOc1BjUVZKOFlUY1NjcUE9PSIsInZhbHVlIjoiay9GV2VDOTlPUjB0Tnp2anZzeVMwRWNRRGcxUFA2aVIwcFIvNDRoZ0d2QjBhYXRnenFxOVRzbDM2b21ZamJJUE9uQ3BFQUpjOWtTaCt6b09tMmN2MmJ5MDhUZFhNZ2pMZUpiRUV6eFU1ejNHOCtTVENQN04rUVZxZDhLTU92bVIiLCJtYWMiOiI1NjI3YWM2MzU0NjQwNTY5ZTZjNWIzOTM3Mjk4NzJjZjBhYWJiZWZhNGI5Njc1YzRmYjcwNThiOWRjMDFjNjliIiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6IklLb3dvN2NOc1BjUVZKOFlUY1NjcUE9PSIsInZhbHVlIjoiay9GV2VDOTlPUjB0Tnp2anZzeVMwRWNRRGcxUFA2aVIwcFIvNDRoZ0d2QjBhYXRnenFxOVRzbDM" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InNHSk0veXVPUEw0c2tScUN4V2szN0E9PSIsInZhbHVlIjoiSkJkQmZrQmdLY3dFZGZzYk1BanpvK0RERGtMSEFFSmY5Zlk1OE9sTHE5NFZRREp2VER4YlRsLzMxQUNNd3VNRFBVbE4xM3VPamVXcmJ2NDZGYndlRDczZVlsenA4bDR0dEtFekUrcHFlQS92Unh4SldFZi8xM25qOFhZLzNkMUsiLCJtYWMiOiI5N2FlYThlODZlNTVhNGMyYmEwZGE0MmQ5MzliZGIxMDk4N2VkZDg2ZmQzZmE5MGIxMGExZTU2YTg4MjFmOGY1IiwidGFnIjoiIn0%3D; expires=Thu, 20-Feb-2025 11:23:21 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6InNHSk0veXVPUEw0c2tScUN4V2szN0E9PSIsInZhbHVlIjoiSkJkQmZrQmdLY3dFZGZzYk1BanpvK0RERGtMSEFFSmY5Zlk1OE9sTHE5NFZRREp2VER4YlRsLzMxQUNNd3VNRFBVbE4xM" 1 => "alphaeurekaselectnet_session=eyJpdiI6IklLb3dvN2NOc1BjUVZKOFlUY1NjcUE9PSIsInZhbHVlIjoiay9GV2VDOTlPUjB0Tnp2anZzeVMwRWNRRGcxUFA2aVIwcFIvNDRoZ0d2QjBhYXRnenFxOVRzbDM2b21ZamJJUE9uQ3BFQUpjOWtTaCt6b09tMmN2MmJ5MDhUZFhNZ2pMZUpiRUV6eFU1ejNHOCtTVENQN04rUVZxZDhLTU92bVIiLCJtYWMiOiI1NjI3YWM2MzU0NjQwNTY5ZTZjNWIzOTM3Mjk4NzJjZjBhYWJiZWZhNGI5Njc1YzRmYjcwNThiOWRjMDFjNjliIiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6IklLb3dvN2NOc1BjUVZKOFlUY1NjcUE9PSIsInZhbHVlIjoiay9GV2VDOTlPUjB0Tnp2anZzeVMwRWNRRGcxUFA2aVIwcFIvNDRoZ0d2QjBhYXRnenFxOVRzbDM" ] ]
        session_attributes
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        array:5 [ "_token" => "e8pyG5P6LGCEJMiadc63335KljUP82PdFbbB474d" "uc_cart_id" => "bd56c085dafe9b4ae0726412601516fc" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/930" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-17 23:23:21GET/chapter/930112189